Oy- can’t blame this one on Godaddy after all. It was a misspelling- a missing ‘y’ in the specification of the domain name for the cookie. I just tested w/the regular cp rather than the admin.php file(s)- but that was the core issue. Can login using sessions+cookies to the backend and flip between sites no problem there. May/may not need to tweak admin/path files depending on what all’s been done. But core issue should be solved.
Test her out- let us know how it goes.