@congthanhgiong thanks for pointing us in the right direction.
It turns out that we had forgotten about some custom HTML we had put into some grouped products. In an effort to have a "buy" button at the top of the page, we had some custom anchor tags in the short product descriptions that would jump you down to the bottom of the page where you could choose your option(s).
Anyway, we re-worked those and that seems to have solved the problem. I guess in the previous theme, they didn't cause a problem, but in this one, they did.
I appreciate everyone that looked at this (whether you commented or not). The problem was, of course, our fault. :P