do I need to replace the entire faucet or just install a handle? It looks corroded inside.
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In general, if you are missing some parts of a faucet or valve, you will have to replace the whole thing. You should not expect the core from one faucet to fit the body of another. Each manufacturer, usually for each model, makes the body, core screw, compression nut, handle, etc. to fit together in a unique way.
The only dimensional standard is the soft washer and the seat that it presses against, and even these are made in several sizes and materials.
Sometimes you can get lucky and find in a hardware store a new faucet of the exact same make and model as the one you're trying to fix. Then you can take it apart and use some of the pieces for your repair, which saves the labor of replacing the entire thing. But you're not going to be able to buy parts for a faucet or valve, except for the rubber washer.
Looks like it's time to change it out to me.