Have you or the previous owner(s) ever had the windshield replaced? If so, many glass techs have not had much experience on replacing the windshield on a XLR or a C6 Corvette. Hence they are unfamiliar with removing the rear view mirror correctly.....and by doing it wrong causes the internal pins to be bent or broken. This is what causes the rear view mirror to wobble.....being very frustrating to the driver. The correct way to remove the mirror is to turn it 1/4 turn counter-clock ways to remove it w/o bending or breaking the pins inside of it.
If this is your problem you can check at a GM part counter to see if these pins are still available to order, or in stock >> #25896360 .. 3 new pins are advisable to work with. There is an optional way by buying 'drill bits' but takes more work to accomplish the job.
I'd suggest that you watch the video on You Tube carefully to learn how to D-I-Y >>>
For those owners in need of a windshield replacement, I'd search out a good glass tech who has done many Corvette windshields and ASK THEM
if they are familiar with removing the rear view mirror w/o bending or breaking the pins. Ask them which direction to turn the mirror to remove it...
if they can't answer that question, you need to find another tech to do the job.
Some unfamiliar techs will cause the wobble problem and not tell you it was their fault.
I'm actually considering a replacement myself, due to the same issue. Mirror is loose at the adjusting joint, not the windshield contact area. There are 4 tabs that can easily break off, and I've already lost two there.
Is there a brand that would be compatible with the OEM 16-pin connector?