I don't know if I'd so much call it "recovery." That makes it seem like anxiety is something you have or don't have. I feel like my anxiety is part of my personality and it doesn't really have an on/off switch, it's on a dimmer. Can anxiety be dimmed to the point where you're not symptomatic? Absolutely.
I've always had a very anxious personality, but about three and a half years ago, I started having panic attacks and became very health anxious. My doctor prescribed Xanax and suggested I see a therapist. I did find one, and even though I had insurance that covered mental health at the time, I chose instead to go to a school of psychology where the therapists were students in their final year, earning their clinical hours. The only difference between my therapist and and one offered through my health insurance was that she was not licensed—but big deal. I paid out of pocket, but they did charge on a sliding scale and did not even accept insurance. Still, sessions were $60 per hour, and it WAS a sacrifice of both money and time, but it was worth it. I don't know where you live (I live in Los Angeles), but if you can find a school of psychology, it might solve your insurance catch-22.
There were two really important things I learned during that time that helped me get better:
One was learning that all of the physical anxiety symptoms I had were exactly that—symptoms of anxiety, not some deadly disease, and that by reacting fearfully (by jumping onto google, by running to the doctor, but interpreting everything I felt in my body as something sinister), I was only prolonging the problem. I think very often we just really want that unpleasant symptom to go away and THEN we'll stop worrying about it, but the opposite is actually true—we have to stop worrying about it so much and then it'll slowly go away—it's literally a matter of, when you feel that scary sensation to NOT react over it. Is that a natural feeling? No, but it gets easier the more you do it. "Getting over" health anxiety is just about getting your stress levels down to where the body is no longer reactive. From the time of my first ER visit to the last time I ran to the doctor in fear was about nine months. I still get the occasional symptom but I know now that it's just stress.
The second was learning that being scared/preoccupied/obsessed about my health or what my body was doing/feeling was really just a distraction from all the other stuff in my life that I didn't want to deal with. And I'm not talking about conscious avoidance, I'm talking on a subconscious level. Freaking out about possible heart problems or liver failure or ulcers (or leukemia or lung disease or a blood clot or ... ), though unpleasant, was much easier than facing my money worries, or uncertainty about the future, or career disappointments, or feelings of loneliness. Getting to the bottom of this has been a much longer process; something I'm still working on. Even though I don't fixate on the state of my health any longer, I still tend to fixate on other things; I still engage in worst case scenario thinking and am pretty pessimistic. Being miserable and negative are attitudes that have been ingrained in my personality for a very long time, so this entrenched anxiety is taking longer to sort out.
I no longer see my therapist (I couldn't afford it anymore and she moved to Texas, anyway), but I do meet with a nutritionist once a month online who specializes in clients with anxiety disorders. She recently recommended a book called "When Misery is Company." I haven't started it yet, but it might be helpful to you. I also found a particular online anxiety website, that offers the bulk of its content for a small monthly fee (like under $10 a month), to be EXTREMELY helpful when my anxiety was at its worst. If you want the address, I can send it to you through a PM, but since it's a commercial site, I'm not allowed to include it in a post. Anyway, let me know.
But yes, long story short, getting over health anxiety IS possible, for sure. Lots of people do it without therapy, it's just a matter of finding out what works best for you.