Perhaps I'm confused about your problem. I thought you had a leak where
the stub out pipe and the trap were joined. I think that only the trap
contains water. The pipes on the sink side and the stub out side should
drain and should not hold water. If what I've just described is your problem
then let the joint dry for a day (don't use the sink). It should be bone dry
if no one uses the sink. Caulk the joint and let the caulk dry for a day.
Spread some newspaper under the joint and begin using the joint. Look for