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False alarm. Right after I posted this, I went back to working on the site, and it started happening again.
I've worked on the config and figuring out the system for a few days now. How can it all of a sudden start doing this? Doesn't make sense to me. It all but makes it unuseable.
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My troubles started after installing the 'Theme Changer' module. That's when I started getting the admin looping condition. Everything had worked fine until I installed that module.
I removed that module, and everything seems to be working okay now. Is there possibly something in that code that's creating this condition?
I'm not versed in php coding, but it seems hard to believe that could do it. But for now, the Theme changer is not part of my site.