I got a 120VAC to 12VAC 3A center-tapped transformer.
12V 3A should give way more than 7V! The 7805 can't provide 3A, so it's very unlikely the transformer is being overloaded.
Are you connecting the center tap to gnd?

You should see 100/120Hz ripple, not 50/60, otherwise one of the diodes might be KO.
Edit: Oh! Is it 12+12VAC? Or 12VAC between ends?
This would explain the issue, as it could be 6+6VAC.
Measure VAC between ends of the tranformer, and also between each end and the center tap.
I'm getting the feeling I should just use a 12VAC wall wart at this point. Variac seemed like a good idea at the time.
Yeah, any cheap 9VDC 3A adapter and it'll be fine, or just dump everything and use a 5V 3A adapter directly!