Try GE Silicone II (Black) Interior and Exterior Window Caulk (usually in the paint section of Depot). Spread it with a 3" flexible putty knife on the affected area. I am going on 7 years with my cockpit carpeting with no issues.
Since most original backing is intended to pass through excess water, now the water will take longer to dry, if you coat the entire back. But, for dripping wet swimmers amount of water, I did not notice any longer time to dry. And no mold issues.
Saves a lot of money in not having to replace.
Joe