It is hard to see where the water comes in. Put the plug in the boat and tilt the trailer to the rear, then fill the bilge and see where it comes out. If the transom is soft it will leak around the plate, if it comes out of the exhaust relief holes the ehaust tube is loose or has a corroded hole in it. The exhaust bellows cannot leak but the shift or u-joint bellows can leak. Another possible leak is a hole corroded right through the transom plate. Most will reverse leak and show themselves.
good luck