Ok, I figured out I can edit the XML file and remove the following line from the top of the file.
<SwisConnectInfo Server="ServerName" ServerType="Orion" UserName="ServerName\UserName" Password="nyUvci0ua5awDQTy8LKdPg==" />
The password field is a hash of some sort of the original password, but by removing this line from the XML, then any user that can log into the Schedule Unmanage Utility can open that job without error. When saved, it puts the above line back into the XML file using the current users credentials. I guess I'll have to try and recreate the global local admin account and hope solarwinds doesn't drop its "manage permissions" again.