I also just bought a boat that kept wanting to sink on me,
basically you have to get under there and carefully inspect for loose rivets you can just move with your hands.
Then once youve found your problem spots clean them up as much as you can, sand them, clean again, and apply JB water weld ( a tube of putty you just knead to activate) let it dry solid and that should stop your leaking problem. Its a long process but it ends up being worth it, my boat stays bone dry now and couldnt be happier. Oh I also sprayed a stop leak sealant over everything once to insure the putty holds.
good luck!