You've had it since 1989 and not once cleaned and greased it?
Don't even give it a second thought, open that thing up. Looking at the parts diagram, I don't see anything crazy, it looks similar to an Abu Garcia Ambassadeur as far as the gear side goes. I see a few possible failure points. The first is the spring washers for the drag may have broken and are jammed. This seems unlikely, but who knows. There is also a spacer between the springs and drag washers, which I'm assuming is plastic. I suppose that could break. I think it's also very likely your problem is not with the drag at all, and that your are actually stuck in free spool. Possibly from 35 year old grease, now more like peanut butter, is keeping the mechanism from returning and allowing the gear to engage the spool. Or maybe it's none of those things. It's a reel, its fixable.
If you really want to fish that reel, I would for sure get new spool bearings if yours were making noise. You can oil them, but why take the risk of loosing casting distance. The other thing is you will want carbon fiber drag washers. Theres no real reason not to use them, they are way better than the felt and teflon washers of 1989. A small tin of Cals drag grease is what you want for both the drag and gears of the reel.