First off, GSM wont move mailboxes in Exchange for you - it moves Unity subscribers from one Unity server to another - in your case the moved subscribers will still point to the same Exchange mailbox they did when they lived on the original server. You'll have to do this in two steps - move the Unity user and then move their mailbox in Exchange to be on the mailstore you want. Unity will pick up the new mailbox location the first time the user calls in to check messages.
Second, the easiest way to make sure the account has proper DB access on both Unity servers is to add the user in question to the local admins group - this has full DB rights for UnityDb. If you want to be more granular than that, check out the "Programatic Administration of Unity" paper on the Documents page of www.CiscoUnityTools.com - it goes into detail how to do this, including screen shots of the SQL adminsitration console. Should get you what you need.
