Although I don't really know what caulking Jayco used on your trailer, I'd like to float an idea. In my trailer, Jayco set the kitchen and bathroom sinks into a putty like material. Over time, that putty material squeezes out. If this is the case with your sink, you might consider using a plastic scraper - which you could get in the paint department at a home store - and scrape off the excess putty.
I did this for our kitchen sink, and then caulked around it with a white silicone caulk. The silicone would probably keep the putty under control, and would end the stickiness.
There's lots of advice and information in forums... sometimes it is correct. For example, all of my posts are made by a political appointee who got the job as a reward for contributions to my diesel bill.
2011 Jayco 28.5RLS; 2008 Chevy Duramax; Pullrite Superglide Hitch