Finally got around to trying your suggestion...
I actually removed the entire sp web application containing both sites (intranet and extranet) and started over. Unfortunately, I'm still hitting the same goofy problem.
I did try another experiment. If you remember, I had to environments and one of them works but not the other. I changed the FS info in the web.config for the environment that works to point to the FS for the environment that does not work. Doing some testing, it turns out that pointing to that FS causes the other environment's people picker to fail as well. To me, that sounds like the problem is in the FS, not SP. That is good news, because SP is a beast.
I'm going to poke around with FS to see what I can find. I'm considering deleting the trust relationship and re-establishing it. Any other ideas??
Thanks again!