At this time, you may also find that some internal parts are worn or corroded. Clean them as best you can or replace them. Clean and polish, or refinish. Most old fixtures can use a cleanup, and now's the ideal time to do it. First, figure out what the parts are made of.
Solid brass cleans up nicely with a solvent such as nonflammable chemical stripper.
Wearing heavy rubber gloves, brush or sponge on solvent. This is best done over a plastic tub. Of, if you prefer, just let the pieces soak in the tub, as directed in the solvent manufacturer's instructions. Using a sponge, gently wipe away residual finish or corrosion. Attack stubborn spots with 0000 steel wool.