We had a similar thing happening with our rescue Maggy but on top of drooling, she tries to get herself under my DH's feet while he is driving. We put a board to block her from between the two front seats and put her Thunder Shirt on her. She weighs about 85 lbs and this seems to help her relax a bit. We have had her for more than 2 years now and have been across Canada and back and forth to AZ several times so I had hoped that she would finally become comfortable but the fact that being a rescue, it is difficult to understand what her life was previously so we just try and do the best we can for her when we travel.
Thank you for giving a rescue a second chance.