Nah, it's definitely not an indexing issue.  I've tried re-indexing several times, and even cleared the search tables beforehand.  I know that everything is working on the DB end, because I've run the stored procedure "GetSearchResults" directly and it comes back with exactly what I'd expect (different results according to search terms).  I have a feeling the problem lies in the Search Results module.

I'm logged in a host. 

And the problem's not that I don't get results back.  For example, if I search for "this phrase", I will get back appropriate results.  If I then seach for "that phrase" it still gives me the results for "this phrase".  No matter what I subsequently search for, I will continue to see the results for "this phrase" until I close my browser and start again.  If I reverse this and type "that phrase" first, I will get an appropriate result set, but then I can't subsequently get the results for "this phrase".  So it's not related to access, since I do see results for either phrase depending on which one I try first. 

Seems like a session state issue or something.  I guess what it boils down to is, the search works perfectly, but only the first time I use it. :(