there is nothing economical about this task. I would experiment with water to try and determine specifically where the leaks are. Chocks, stanchions and stays are often the culprit since they have more forces applied. If you convince yourself that the rail is the problem, the usual task is to lift the rail, rebed and refasten. The leak is usually coming thru bolts holes. The deck/hull joint itself may (hopefully) be fine. I do not know the construction detail of this joint on a Morgan. You should find out how they built the boat and specifically this joint.
I would not replace the teak rail if it is sound. I have also loosened rails without removing completely, rebedded and then set. The condition of the fasteners and your access to them will determine if they are reusable. If you have to remove the rail, replace all the stainless. A boat this age is due for this sort of work. My 78 Bristol has one rail done and one to go - a few small drips when we bury the rail. I use a moisture meter to monitor my decks and Bristols have great access (no liner) so I can see whats up in most locations.
Good luck - big job ! Larry