Title pretty much says it all guys.
actually your title leaves about 50 questions.
Last time our boat was used the engine didn't want to turn over, with the drive down. Actually almost melted a battery terminal out of the battery. Since then the boat has just sat for about 6 years.
you and your dad should have done some diagnosis 6 years ago. there are many things that could cause a battery terminal to melt, including a dirty terminal. especially if the battery terminal is dirty, then only the battery terminal will melt. if the terminals were clean, normally the whole battery cable will melt.
what diagnosis have you done?
I sort of remember reading somewhere the issue is because the bearings in the upper gear case have essentially welded themselves because they got too hot.
this says you dont know because you read it somewhere doesnt make it true for your case. you need to start from the basics. first, clean all your battery terminals. pull the drive (should take only about 20 minutes). do not damage the shift cable. follow the manual on removing the shift cable. that little beauty is now $750. you need to determine what is actually wrong. if the troubleshooting narrows it down to the drive, you need to either have all the right drive tools and take your drive apart to determine what is wrong, or take it to a qualified OMC mechanic. which may be difficult, because you are looking at best for a retired guy that does this as a hobby.
to weld the bearings, that means there was no oil in the drive. I have witnessed it, and it required a crack in the water pump housing to pump the gear oil out and water into the upper on a stringer.
if it only turns over with the drive up, your ball gears are also shot. those get replaced every few years. your gear oil should be checked every few months and replaced every year.
your 78 should be first year of the hydro-mechanical shift, however could also be last year of the electric shift. need to actually check what you have.
hydro-mechanical shift only lasted 3 years.
