I had a similar problem on my EE with Mapdroyd, but was able to reinstall the older version and say no to offered updates. I don't know what the problem is with Sugarsync, but I sincerely hope you can get it back to working!
Are you running (any version of) Ermine, or are you on Dingo?
Edit: I just checked the Amazon Appstore. They have version 3.3.3 which looks like it was last updated in July. If your version is later than that, and if you have access to Amazon, you might want to try their download. It requires Android 1.5, so should work with either Dingo or Ermine.