My dog won’t stop licking her paws

My dog won’t stop licking her paws. They are now all swollen, red and skin is broken on 2.
It started about 6 months ago. The vet said it was anxiety and gave her ‘Prozac’
It didn’t help.
He then said it was an infection, put her on antibiotics. After the first months supply of antibiotics there was no improvement. He did some tests and said she had strep and a staph infection and gave her more antibiotics.
She’s been on antibiotics for over 4 months now, no improvement; it just keeps getting worse.
I’m sure most people can agree that vets are not cheap and with almost 6 months worth of medications and several visits, I was hoping someone could give some advice thats maybe been through something similar; so I could possibly give the vet some ideas on what our next steps could be when we go back this week.
Also, I already changed her food to a grain-free, hypo-allergenic; also no improvement.
I’ve read a lot about soaking dogs feet in apple cider vinegar & water but I think that would hurt/sting her, as she has open sores on 2 paws now.
Any help would be super appreciated.
My dog is a pitbull and I’ve had bad luck in the past with other vets; this guy is super nice to her so I was hoping not to switch vets but 6 months with no results and a lot of money spent is getting pretty discouraging.
Thanks so much!!!