Nice boat, I always liked the starwinds, solid designs for lake/trailer sailors.
It sounds to me that your cockpit scupper hosts or through hulls are almost certainly toast. On most trailer boats they are plastic thru hulls. Get a couple stainless steel ones, some 5200, a couple new bits of hose and clamps, done.
The idea that your cockpit isn't going to get wet and it will be fine draining into the bilge is pretty absurd in my opinion (no offense) and frankly dangerous. Why would you have a large open surface draining *into* your boat. Boats are meant to keep water out. Just fix them correctly and then you don't have to worry about what ifs (what if it rains while I am out, what if my cover leaks on the trailer, what if my bilge pump breaks or gets clogged.
Plugging the sink isn't my style since I would again just fix it if it was broken. How are you going to "plug" it? Literally a plug in the drain hose? Stuck in the thru hull? Still got a hole in the boat and no reason your plug won't fail like anything else. Best fix is a seacock you can close unless you need it. About $80 gets you a proper marelon seacok/thru hull one and wouldn't take more than a half hour to install. https://www.fisheriessupply.com/plum...s-and-seacocks
Good luck and happy sailing!