1) Change your password immediately and virus-scan your computer.
2) Keep a tally of all your items. Collect all your rares and gold cheques, count them, write it down.
3) Check them as soon as you log on every day. If something's missing, notify a GM that your account has probably been hacked and items stolen. (Not that they'll do anything, but it's better to be safe than sorry.)
4) If things go missing, you're being exploited. Put UO on a USB stick or something, find somebody else's computer, and change your password there; then take your own computer in to a professional to have it thoroughly checked out for spyware, if the virus scan didn't catch anything.
This is very much erring on the side of caution, but if your characters keep shifting around and you're getting worried about it, consider going through these steps. Good luck!
Edit: Okay, I missed your second post about using a different computer each time. But if they're key-logging you, they'll still be able to get your changed password as soon as you enter it on the infected machine, so you need to change it and then - without logging in once after you've changed it - take steps to clean your computer out.