Dave, you're a genius! I think that might be the problem.
The old host used
http://www.mindplacesupport.com/forum and the new site is at
http://mindplacesupport.com/forum. I didn't even notice!
Changing the base URL in my settings has brought back the "Invalid Username or Password" dialog.
I'll try that for a bit and see if that fixes the problem.
Thank you again!