Pope's Pioneers of Massachussetts also puts him in Charlestown in 1637.
I wonder if it would be possible to find the document that Wyman and Pope are referencing? There were are two other John Andrews in the Directory, one in Ipswich (with a different wife), and the other in Kittery. I wonder if either was a cooper, as that seems to be a helpful identifying factor.
There are some (not very many) people that Anderson missed, so it would not be without precedent.