The faucet should have "seats" that the washer presses, or seats, up against. These will need to be changed. If you can't get them I'm afraid to say you're SOL and need a new faucet.
For something that old your best bet would be a plumbing wholesaler or phone American Standard directly.
We might need a picture, because all 50 year old A/S faucets have replaceable seats. And if you have Re-Nu seats, which is very possible, you have to be careful that you do not damage their sealing surfaces inside the body.