In 81' i could run like a Deer. Not so much 39 years latter, walking is tough today. Your boat is very old and the rivets are leaking. Tring to determine exactly which exact vivets are leaking by looking into the bilge is next to impossible. Plus it many be leaking through the transom, that is much easier to determine. It may be both transom and rivets leaking. Do a transom test. I don't want to be the Grim Reaper but the old girl is 39 years old no matter what. 3 or 4 grand of materials around where I am as well as a few hundred hours of labour will get that hull in new shape. Hope you got it cheap.
Also never, ever fill your boat with water while it sits on a trailer level. Maybe if you lift the bow as high as you can get it to see if the bilge is leaking is OK, once. Take the transom saver off, tilt the engine up 100%, lift the trailer to the max until the skeg hits the ground then put some water into the bilge area. Watch for transom leaks. Best of luck.