Welcome to the forum and thank you for sharing your photos and situation.
Your skin problem looks like the classic typical problem that this forum is all about, so you’re on the right place.
I don’t know much about the costs of Soolantra in the US (I assume?). Maybe @misfit108 can tell you something about it? If I remember correctly, someone said something like $200 for a tube of Soolantra. But you can use it for like 3 to 6 months.
If your dermatologist doesn’t want to provide Soolantra, than you could try to show him this: What does it look like? (photos) and tell him/her that a lot of people on this forum have the exact same unknown skin problem and a lot of them used Soolantra with success.
Soolantra is a moisturizer containing the medicine ivermectin, the primary/active ingredient. Some people have just ivermectin from a topical animal/horse product with success. It can be much cheaper (and without prescription), although it may be safer to use Soolantra. For more info, see: Does yours spread? and Cattle medicine is more potent
Topical ivermectine may cause a bad reaction on some people, which is supposed to be a temporary breakout. So you may want to start with it slowly. For more info, see Bad first reaction
I have also used doxycycline for like 1 to 3 months, but it didn’t do anything to the lesions.
Your skin also looks dry/prone. Dry skin may be more prone to the problem than hydrated skin. So you may also want to try: Skin hydration
(It probably isn’t going to remove the problem, but this moisturizer could reduce itch and reduce the occurence of new lesions a bit.)
Can I post your photos in this topic? What does it look like? (photos)
So that people can compare it with their skin problem?