Is Mayo Clinics close to where you live? Have you considered a University Medical Center in the state where you live? I went to the University Hospital in my state and there didn't seem to be a shortage of rheumatologists. I live in a rural state so statewide there is a shortage.

I remember when I was first diagnosed with PMR ... that was 18 years ago. I saw a primary care provider (PCP) first but I was quickly referred to a rheumatologist. Sometimes a PCP doctor referring you to a rheumatologist carries additional weight than a self referral. If your PCP thinks your PMR is serious you might be seen by a rheumatologist sooner. Even so, expect to wait 3 months or longer to be seen.

I have seen so many rheumatologists over the years. I didn't ever "doctor shop" and I never requested a second opinion. I saw three rheumatologists when I was first diagnosed with PMR. I was self medicating and treating myself until a PCP informed me that I had a "serious problem."

I still can recall the look of disbelief when I told a PCP that the only thing I wanted from her was a prescription for prednisone. I had to explain why I was taking prednisone that was legitimately prescribed by my ophthalmologist for recurring uveitis. I went on to describe all the pain prednisone relieved even when uveitis was in remission.
https://www.mayoclinic.org/diseases-conditions/uveitis/symptoms-causes/syc-20378734
The PCP wrote a prednisone prescription for a few days but insisted that I see a rheumatologist.

A University Hospital has teaching as a mission. You will likely see doctors who are being trained as a rheumatology specialist but are supervised by an experienced rheumatologist. My visits involved extra time for training. I usually saw two doctors who were interested in me and rheumatology.