Jeez Roger, what a pain. And often its something simple
I wish I could help but I don't have gemini. You've checked the obvious I gather? Right mount selected, gemini hasn't defaulted back to its factory setting? The usual stuff?
The only thing I can offer is; we were having trouble with the G11 at the Maplton Observatory, it was doing really weird things, and not consistantly wierd. Like loosing location, wrong time, not slewing to the correct coordinates, defaulting to a different mount at start up. What it turned out to be was the internal battery was shot (in fact it was the wrong battery, and it had been installed in the factory, we had never changed it previously). Change the battery and we haven't had a problem since.
That might be a place to start. How about trying a very quick, hard and dirty, half hour say, dirft alignment and then use the build in corrections? See if that makes a difference.