If the parts inside a friction shifter are rusty, should I replace them?

I recently got a bike from my neighbor, and it has shifting problems. It won’t shift out of the lowest gear on the rear cassette. The shifters don’t do anything. I can pull the cable by hand, and it’ll shift gears, but then it’ll just go back to the smallest gear. I checked inside the shifters, and it looks slightly rusty, which makes sense because it was left outside. Should I get those shifters replaced?