In THAT case, I'd remove the wood, lock back the slide, and use aerosol brake cleaner, pipe cleaners, Q tips,chunks of muslin sheet wrapped around a screwdriver, etc, to get all the crud out of the action possible. Is the bore fouled, too? If so, need bore cleaner, proper sized brush, rod, patches, as with any gun. Relube with something decent, like Lubriplate.
Auto parts cleaner-solvent, air-nozzle would be nice, of course. Lots of clean shop rags, etc. Dont break your back on something that's making you no money, and you'll never need to do again (most likely).
There's rust removers available,ya know. The shipyards always used some sort of "naval jelly", which is doubtless still sold. very light "carding" with 00 steel wool is often helpful in removing rust, too. Yeah, sounds like you'd better take the thing apart, cause rust is most likely where it can't be seen, with the gun assembled.