lots more can be said about this, especially given how new it is, but I've given my analysis on the assumption that it's genuine here and u/y_sengaku has given a more conservative analysis, including the risk that it is a forgery, here.

Both of us are in agreement, though, that knowledge of Markland is not the same as knowledge of "America". While the text is ambiguous as to the size or extent of Markland, it clearly is someplace nestled deep in the subarctic, and not a system of continents. Just because our best-guess as to where "Markland" was is the coast of Labrador, which happens to be part of Continental North America, does not mean that anyone at all in the Middle Ages recognized it as such, conceived of it as such, or attempted to mount an expedition to explore it as if it were. Instead, it actually fits fairly standardly into medieval ideas of north atlantic islands being liminal and full of monsters - the Navigatio Brendani and, actually, The Saga of Eirik the Red are useful comparison's to Galvano's account of markland as a land of giants (the saga has Sciapods, a one-footed people described by Pliny the Elder as living in Ethiopia, in the region).

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Anyway, on the other half of the question - not far at all. If we use the sagas, particularly the Saga of Greenlanders, as the main source, it would seem that the various Norse expeditions went south from leifsbuðir to the mouth of a river, sailed slightly up it, and then met indigenous people. They traded with them at first, and then for uncertain reasons (lactose poisoning has been proposed, since sheep's cheese was one of the things traded) were attacked and had to abandon the lower settlement. This is somewhat often interpreted as reaching the Gulf of St. Lawrence.

In an archaeological approach, we have one confirmed Norse site - L'anse Aux Meadows, in northern Newfoundland. However, ecoarchaeological excavations keep finding material that doesn't correspond to historical evidence, such as this study from 2003 that suggests more significant voyages to the St. Lawrence area or farther south to pick up butternuts, or this study from 2019 that proposes human-deposited ecological material in LAM that dates to the 12th century, up to 195 years after Norse settlement in Vinland is said in the saga material to have ended. Even they don't suggest that it is continuous habitation, though, but rather extremely short-lived intermittent visits - potentially these are due to lumber harvesting voyages, like the Icelandic Skalholts annall says occurred in Markland in 1347.

Both approaches agree with regards to your question, though - these are short-lived settlements on the extreme coast. There is no evidence to suggest widespread inland exploration of coastal Canada, or sufficiently far voyages to reach the modern United States or to understand the scale of what "Vinland" was geologically attached to.