It's strange to have the dishwasher going into it's own trap instead of
the easy way, which is to connect it to the disposal. If the
dishwasher drain is just stuck in the end of the trap, which I imagine
it is, then it leaves an easy relatively low spot for the drain to
overflow from and make a mess. I'd re-plumb it the normal way.

If you want to find out where the water is coming from, take out
everything from under the sink and lay down some paper towels. Check
it frequently and you should be able to spot the area where the water
is first coming from.