The name Dollery originated from France and came to England around the 1066 invasion.
Yes there is some proof that the Dollery's name sake did settle around Hampshire and later spread from there towards London, then headed to Portsmith. The first one to enter in to America in the 1600's was an Anthony Dollery, then in to Canada, Australia, etc.
I do have Dollery Birth and Passing records dating back to 1500's.
IF you require any of that information we can discuss what you may want to see.