I have had this problem with my 1983 - 323 and it was not a difficult job to repair.
I am not familiar with other Morgans. What I did with mine was remove the stainless steel rub rail. This is what covers the joint between the deck and haul on my boat. I got a grout cleaning brush from HD and cleaned the joint with the brush, this helped remove any loose old chaulking. I used denatured alcohol to clean the joint for better adhesion. I then used 100% silicon that I purchased from HD and applied this to the entire joint.
I seen no need to remove the teak since this was screwed to the deck and severed no purpose other than for looks. I replaced the screws that held the rub rail with ones that where a 1/4" longer because some of the old ones where not tight.
The caulking material just gave out after 25 years. I only did this to the sections where there were leaks.