Thanks for the link to the PNAS paper. From what I could tell from an admittedly quick read of the article, though, it makes no claims about a "300 year drought" during the medieval period in North America. What it does say is that drought events were common during this time period, and that they often persisted for one or more decades. For example, the article says, "the 12th century medieval drought persisted with an extent and severity displayed in the worst-case decade, 1146–1155, for two decades, 1140–1159". That's not 300 years of continuous drought. Yes, the overall mean precipitation in the Southwest was lower during the medieval centuries, but that doesn't mean there was continuous drought during this time.
For what it's worth, the conclusions of the PNAS paper are pretty much in total agreement with what the researchers in the NASA study found.