So what's the truth?

The truth / historical fact is that by the time Europeans reached North America, plant domestication / agriculture had a history several thousand years deep, and the majority of the population from the Atlantic coast to west of the Mississippi River was living in permanent villages, towns, and urban centers supported primarily by food production.

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When Europeans arrived in Eastern and Southeastern North America in the 16th and 17th centuries, they encountered fully agricultural cultures-- civilizations, if you want to apply that somewhat baggage-laden term. These cultures mostly farmed the famous three sisters: corn, beans, and squash. Interestingly, two of these crops (corn and beans) originated in southern North America (Central America) and in South America, and were brought / imported north over the last 5000 years.

Southwestern Native American cultures were growing corn 4000 years ago, and by approximately 2300 years ago (plus or minus a couple centuries), corn appears to have been incorporated into the crop complex grown in the Eastern Woodlands. However-- and this is important-- these were not the first crops to be grown in Eastern / Southeastern North America. Instead, corn and beans were folded into an existing set of indigenous agricultural practices that had been developing in North America since at least 5,000 years ago, and arose out of earlier practices dating at least another two or three thousand years earlier.

Middle (8900 to 5700 BP) and Late Archaic (ca. 5700 to 3200 BP) indigenous Eastern North American cultures began experimenting with plant husbandry / cultivation (horticulture) at least 6000 to 8000 years ago. It is strongly suspected that care and management of productive stands of masting trees, for example, may have played a role in the development of ideas about plant cultivation, mass processing of resources, etc.

The first conclusive evidence we have of early efforts to cultivate (and ultimately domesticate) plants in North America is in squash and gourds. These presaged increased cultivation and experimentation with a variety of other seed-producing plants beginning around roughly 4500 to 5000 years ago or a bit earlier. It's likely that the earliest squash and gourd cultivation focused on the edible seeds of the plants rather than the flesh of the plants, although the flesh likely was (or became) also a significant source of food. Domesticated versus wild remains of these plants often hinge on seed size differences (domesticate were larger) and it's very likely that selection for larger seeds had a lot to do with the domestication of squash and gourds. But rather than focus on squash, I want to lean more into the seed-based plant domestication efforts.

Evidence from archaeological sites excavated in Arkansas, Tennessee, Kentucky, and Missouri (especially rockshelters, which are excellent environments for preservation of fragile plant remains) indicates that North American peoples were collecting the wild seeds of oily seed-producing plants such as marsh elder and sunflower and starchy seed-producing plants like goosefoot / chenopodium, erect knotweed, little barley, and maygrass well before they began experimenting with cultivation, some as far back as 7000 years ago. Some of these plants-- chenopodium, sunflower, marsh elder-- with a longer history of collection also appear to be the ones that were domesticated fairly early. Early evidence of domesticated sunflower was recovered in Tennessee at the Hayes Site, dating to around 4300 BP. Earlier domestication of that crop has been proposed at closer to 4800 years BP.

Domesticated chenopodium (a relative of modern-day quinoa) has early dates of around 3800 to 4000 BP, and marsh elder has dates of around 4400 BP.

These crops and others that were added over the ensuing centuries have been referred to by Bruce Smith as the Eastern Agricultural Complex. Some were fully domesticated, meaning that the plants were genetically changed as people selected for preferable traits (size, ease of processing, etc.). Others were only cultivated wild (maygrass, little barley) and were never domesticated. But the development of this agricultural complex in Eastern North America makes the region one of only a relative few in the world (current estimates are around about 8 places) where agriculture / plant domestication was independently invented.

The plants of the EAC together provided a substantial base from which sedentary cultures could expand and flourish. Beginning in the Late Archaic period, we start to see larger and more permanent settlements in river valleys. Monumental architecture-- not necessarily a product purely of agricultural peoples, as we can see with Poverty Point-- expanded significantly with these sedentary societies. The mound building cultures of the Eastern Woodlands flourished. After about 3200 BP, evidence suggests that peoples across much of eastern North America were engaged in significant food production using the crops of the EAC, and much of the population was shifting to sedentary or semi-sedentary ways of life.

By about 2300 years ago, evidence suggests that maize / corn reached the Eastern Woodlands. A Mexican domesticate, the early forms of corn were much different than modern corn, with quite small cobs that produced only a handful of kernels. Nevertheless, corn was highly productive, exceeding the calories per acre of pretty much any of the other indigenous EAC crops. The crop appears to have rapidly displaced most of the indigenous crops. Most archaeologists who study this period and subsequent periods in indigenous American history / "prehistory" attribute the rise of the Middle Woodland Hopewell cultures, and Late Woodland farming cultures that gave rise to the Mississippian civilization, to the high productivity of corn. By the terminal Late Woodland period, ca. AD 800 - 900 (around about 1100 years BP), we see the emergence of a geographically expansive set of cultural and social practices that are broadly described as "Mississippian." Some of the earliest Mississippian centers-- e.g., Cahokia outside St. Louis-- were enormous, featuring colossal earthen mounds constructed with sophisticated soils engineering practices that modern-day earthen dam and levee designers would recognize.

The Mississippian civilization extended from west of the Mississippi River to the Atlantic and Gulf Coast, with likely diplomatic ties and other interaction networks with coastal peoples such as the Calusa (who were able to maintain a largely hunter-gatherer subsistence pattern due to their access to rich marine resources along the Florida coast).

When Europeans arrived in North America, they encountered Mississippian peoples living in permanent and well-organized towns and villages across the landscape, which was extensively populated and managed. These cultures were predominately agricultural, supported (by that time) by mainly maize agriculture, although some of the earlier indigenous North American domesticates were likely also grown at that point.

Of note: sunflowers were domesticated exactly once in human history, probably somewhere in what is today Tennessee or Arkansas.

Here's a good source to start with if you're interested.

Smith, B. D. & Yarnell, R. A. Initial formation of an indigenous crop complex in eastern North America at 3800 B. P. Proc. Natl Acad. Sci. USA 106, 6561–6566 (2009).

This is largely very dependent on which group of indigenous people we're talking about (and particularly in which regions they reside). I unfortunately cannot speak to a great effect about the Wampanoag specifically (this is the tribe whom the pilgrims met) but I do know a decent amount about the Wabanaki tribes of northern New England who, suffice to say, would have been fairly comparable to their neighbors.

For starters, to address what I said about the regions in which they reside, different climates led to different practices/cultures/economies. For example the southern and central American groups, like the Incas absolutely utilized large-scale agriculture. Just a look at the terraces in numerous Incan sites clearly indicates that they not only had agriculture but that they also had civil engineering geared around allowing more efficient agriculture. Now these kinds of terraces and vast farmlands were not common in what is today the Northern United States and that's for a simple reason: the climate is not particularly conducive to agriculture. As a general rule of thumb, the closer to the poles you got the less prominent agriculture was in indigenous practices. The Inuit for example do not utilize agriculture because, frankly, why would they even try?

So did the indigenous people of New England have agriculture? Yes absolutely however, unlike most Europeans, they had primarily semi-permanent settlements. This meant that they would move either when the farmland they were using was depleted or if the growing season had ended and there were more rich hunting/gathering areas to settle. Europeans primarily settled permanently in place and (hopefully) grew all the food they needed throughout the year. The indigenous tribes were typically more flexible (as was needed for the less-than-stellar land and climate of New England) and were not solely dependent on agriculture, generally using hunting/gathering to supplement what they grew. As a result New England indigenous farms were generally much smaller than what most in Europe were used to (or even what would be found in the Incan empire for example). They would grow food during the spring and summer, while also hunting and gathering, and during the winters they sometimes overwinter where they farmed during the growing season or move to a new place with more plentiful wildlife for hunting/gathering.

So in short: yes the indigenous people of the Americas had agriculture and were often actually quite proficient with it, within the limits of their climate/resources (no indigenous pack-animals also provided a fairly large barrier to overcome in terms of large-scale agriculture). For this I was primarily going off of general knowledge for the wider-indigenous peoples and for the New England/Wabanaki practices I was using "The Wabanakis of Maine and the Maritimes: A Resource Book" which can be found here.

I would encourage you to do some reading on Native American tribes, many tribes have their own websites and resources available for free online. In my somewhat limited experience I've found that Indigenous Americans tend to be very misrepresented in the popular consciousness.

You have two excellent answers, and I'd like to add a side note based on Dr. Robin Wall Kimmerer's book Braiding Sweetgrass.

When Europeans arrived in North America, they didn't recognize indigenous agriculture because it looked very different from European agriculture. North American fields had corn, beans, and squash growing in the same fields, which looked messy and "uncivilized" to Europeans, whose crops were planted separately.

This led the Europeans to view the indigenous people as lazy because their fields were small and weedy. Those Europeans didn't realize that maize has a much higher yield per acre than their wheat, and that the "weeds" were other food crops (that actually suppressed weeds in the fields). North American agriculture therefore could be accomplished on smaller areas of land and with less work.