From linguistics we know that, as a general rule, languages that are isolated and lacking recorded form (mostly writing, but also plays, radio, cinema etc.) tend to mutate the most. Isolated tribal languages in the Middle of Nowhere Caucusus Mountains tend to mutate faster than the written language of an empire, because use of that language is usually limited to casual conversation among a small population, and abbreviations, idiosyncrasies and slangs tend to permeate the speaking population quickly.

On the other hand, all the colonies in the New World started with writing and had continuous contact with the Old World even after the area was decolonized. A constant influx of immigrants also kept diluting the particularisms of the Americas. It was only a few hundred years between the first colonies and the industrial revolution which decreased the distance between cities and homogenized the linguistic exposure of all people speaking those languages. The death of most Native American societies through plague and genocide also reduced the effect of local language on the new American dialect.

Additionally, if you are a white, Anglo-Saxon American you might think English is pretty homogenous, but on the fringes of society language becomes more esoteric. The most famous example is African American Vernacular English, commonly known as Ebonics, which due to a history of segregation sounds distinct from standard English. Quebecois French is also famous for having numerous idiosyncracies compared with French French. On the more extreme and lesser known end of the spectrum, numerous creoles such as Haitian Creole, Jamaican Patois and Guianese Creole have developed from a combination of French/English, African languages from slaves and local indigenous dialects. These languages definitely have a family resemblance to English and French but are barely if at all mutually intelligible.

To summarize, it did happen, but not as much as in the middle ages due to the proliferation of writing and increased contact between regions, and the places where it is very evident are less well known.