I have two roof drains, can I seal one of them?
You have a redundant roof drain. If one of them were to fail (clog) the other would (hopefully) continue to allow water to drain from the roof. If you were to cap one of them And the only drain were to become clogged water would accumulate on your roof to possibly it failed (or found a new route to the inside).

as for it leaking: it would be easiest to open an access to the drain piping inside the roof. Start were you said the water is wetting the ceiling and cut out a opening (14x14) to allow you to inspect and/or repair the pipe. The pipe, if it is original from the 1930's (geez, hmm) I believe would be 2-4" threaded galvanized? Maybe 2" cast? It may even be large diameter copper pipe. What ever the pipe material replace any rotted or rusted sections with ABS of comparable diameter. Join the new to old with rubber repair sleeves that tighten with hose clamps.