OK, so, after support finally got back to me, I discovered that there was some sort of loop going on where who ever was hitting my site was using the "www" in front of the domain. This was never a problem on the old server, but armed with this information I was able to fix the problem and now everything seems to be relatively cool.

The only bad thing is that the fix now requires me to do more work when ever I add or remove things to my site layout. After a while, I'll stop doing that for good with a little luck.

Anyway, so everything should be fine, though all of my sites are virtually identical until I have the time to tweak the layouts individually. My kingdom for a cosmic pause button!