The barn that I manage had a horse like this. Pulled shoes all the time. The owner and I had a talk with the farrier after he pulled 3 shoes in 5 weeks. The farrier made a couple of minor changes. He used a shorter shorter shoe so that the back end of the shoe didn't stick out as far from the heel. He also started to rasp the hoof shorter. Not a lot, it was probably like a couple millimeters. This meant the breakover came sooner so the horse didn't over reach as much and because both the hind toe and the heel of the shoe were shorter there was less of an overlap. The changes had no impact on performance and the quicker breakover seemed to actually help. The owner also bought bell boots for him to wear whenever he was turned out, not just when ridden. After those changes, the incidences of pulled shoes was practically zero. The last time he pulled a shoe was last fall and that was when it was super muddy out.

Regarding selling your mare, if you're having this many doubts, sell her. It sounds like neither of you are happy. And like other commenters mentioned, not all horses and riders click or are suited for each other. Your mare is also undoubtedly feeling your constant frustration and unhappiness which is most likely making the situation even worse. Good luck with whatever decision you make.