Ok. I think the worst is over. I haven't figured out how to randomize the link listings, but that's pretty minor. I've fixed up all cases of deprecated functions in my theme. This could have gone better, but it could have been worse. Good thing I'm a coder though. This wasn't simply plug-n-play.
PLUS! I couldn't figure out why my blog looked just fine but all my admin pages were wonky. At first I thought I'd screwed up the installation completely and would have to restore from backup and try again. Then I had a brainwave and tried Safari instead of Firefox, and it looked fine in there. Turns out I'd disabled Java by default in my Firefox browser for security reasons. WordPress didn't like that because apparently it's now heavily Java dependent.
I must say that my hosting provider made my life much easier by upgrading my server out from under me instead of having me move to a new server. That meant no e-mail backups and complicated database restores. Just the WordPress upgrade to do itself. Yay Mecca!