"Leo, there is a gasket insert
"Leo, there is a gasket inserted into a groove all around the transom shield. If the wood gets rotted in way of the transom contact area, the gasket may not stop water from coming in. Also, the transom shield is attached with 6 carriage bolts and nuts to the transom. If the nuts are not tight, there will be water ingress between the transom and the transom shield.
I would start making sure that all 6 nuts are tight. YOu shold be able to reach all of them (with varied grades of difficulty) from inside the boat.
Then, I would have a look at the shift cable sleeve. If it leaks, yoy may be able to temporarily patch it with electrician's tape or similar. Theoretically, you should be able to unscrew the sleeve (hose assembly) and install a new one from inside the boat, provided the boat is on a trailer and you have disconnected the whole cable from the outdrive. And I say "theoretically" because many times the brass fitting will simply refuse to become disconnected from the transom shield and you may have to use heat or drilling.
If the exhaust pipe leaks, then you should be able (again, theoretically) to replace it from inside the boat. Problem is, the exhasut pipe contact area in the transom shield may be pitted (electrolysis), in which case you will have to replace both the transom shield and the exhaust pipe. Personally, I thing this job (replacing the exhaust pipe) should be done with the engine/bellhousing disconnected from the transom shield and the engine/bellhousing removed fromt he boat or at least brought forward out of the way (if there is enough space) to do the job comfortably and to asses the condition of the area."