you should be able to fill your tank right to the top with no leaks! something is wrong for sure!
start by finding the "basement" in your trailer, that is the location where all the piping to the tank goes through the floor. then if you can with a flashlight and possibly a small mirror look to see if the large AND the small hoses from the filler are still attached. you also may need to try to recreate the filling incident while watching this area.
hopefully it is just a hose problem. if not a tank repair or replacement may be in order.
as for drying out the trailer, run the ac unit it will draw the moisture out.
you call them ferrets, i call them weasels.