sorry to say it but I would guess that the back glass seal is leaking. I good glass guy can try to seal it without taking the glass out, but not if the seal is older and not pliable so they can get a gun tip between the glass and the seal. Good news is that a new seal isn't that expensive and you can cure any rust under the rubber when the glass is out. If you are going to trash the existing seal any way, you can cut the seal off the glass and take the glass out yourself, fix any rust and then pay to have it put back in correctly with the correct sealer.
If the trim clips are originals they should have sealer that closes the holes...