Elizabethan times - Ladies with beards...?

Well of course they had razors, even in prehistoric times. Some of the flint knives archaeologists have unearthed are sharper than surgical scalpels today. Not all men had beards, going back almost as far as art exists you can find paintings and sculptures of clean-shaven men. There's no reason to suppose that such unfortunate women could not use the same technology to remove unsightly facial hair.