Is there any medicine that can permanently get rid of oral herpes (cold sores)?

Sadly, there's nothing. Certainly people with persistent genital herpes used to be given suppressive therapy (not sure if this still happens), but it wasn't something that was done very often. Have you ever asked for a referral to a dermatologist? Do you live somewhere sunny, because UV light can flip them out. I'm sure that you're very careful about using separate towels and flannels for your face, as it is possible to pass them 'down below'. The Herpes virus can survive on towels and flannels, so be careful. I'm sure that you know about not having oral/genital contact when you've got them - sorry if that embarrasses you.