I had what was called plantar fascitis for two years. It was worst after sitting down for more than ten minutes, or after a nights sleep. I don't think it was plantar at all, but I finally narrowed it down to some kind of nerve problem, which heals at a very slow pace. For a couple of months, I even resorted to a walking stick first thing in the morning. The docs prescribed Amitriptiline, thinking it could also be Fibromyalgia, but the Ami did help, so it was more likely to be nerve damage.
A few years on, and the pain is now just gone. Completely. Ok, I'll never be able to wear high heels, but that's no big deal. I can walk without pain for most days of the year now, and only suffer if I am wearing shoes that have absolutely no give on the soles.
Suggestions for a diagnosis are all well and good, but you are now faced with what you do now, day to day, when it's at its worst.
To help with the cushioning, if you can afford it, try a pair of Fitflop trainers or sandals and see how you get on. Or even Hotter, or Ecco, although they are not so cushioned. It's worth trying them on in shops to see how you get on. Some expensive trainers have good cushion, but they are mega pricey and not suitable for wearing out and about if you want to look smart.
I started off with crocs, which I hated the look of and didn't support my ankles, then when I found fitflop trainers, I was in heaven. Honestly, I couldn't wear anything else when it was bad, but the cushioned soles let me live my life much more easily while I tried to track down what was going on.
I hope you find out what the source of your pain is. I never did, but am grateful that it does not bother me now, as long as I wear cushioned soled shoes. I'd also be careful of the exercise with that pain. Strenuous exercise could add to the injury if it's a damaged nerve, so go easy on your feet for a while.
Good luck, and I know exactly how you are feeling.