Well, as you can see, I waited a long time before I even tried looking at this. I was waiting to try upgrading to 1.9 until it was stable and until I had time.
Today I tried to piece everything together but I'm stuck. I e-mail Siteground but they don't seem to be offering to do the conversions for free anymore. I'm almost to the point of paying, but not quite yet.
I'm still working on how to do the MYSQLDUMP command from the command line at Siteground. They offer something that looks similar under phpAmin but no character-set option.
If anyone can help me get this working, I'd appreciate it. I'm installing 1.9 to a new directory (uh, I think) just because I'm eager to see how it works, and maybe because a clean install might make it easier to import an old (properly converted?) database.
Thanks to anyone who can help with even a push in the right direction.