I had a sobakawa pillow in the 80s and LOVED it. Unfortunately, it eventually gave me a rash, so I threw it out. I've had some neck issues in the past few years, and bought a variety of pillows to address that. The solution for me was to use a very firm travel neck pillow in conjunction with a pillow filled with millet (so like a buckweat pillow, but with a different grain husks). I sleep on my back (and side) with the travel pillow under my neck, and the millet pillow wedged under the top of my head just enough to keep my head in alignment with my spine. Almost all regular pillows I've found are way too high, and the bend in my neck was making both my arms go numb at night. No more numb arms, and so far, no rash from the millet--it's my favorite pillow (along with my neck pillow).
I got mine from PineTales. No issues with the quality, but I do wish they'd had more of an inner/outer bag construction. I had to wash the pillow fabric once because a pillowcase got left off for a while, and it involved finding a bin to keep the millet in.
Btw, when you get a pillow like that, it will be more full than you need. You're supposed to remove enough buckwheat or millet to make it comfortable for you.
If the buckwheat pillow doesn't work, my physical therapist used a cervical pillow like this:
I liked it a lot in therapy, but I bought a couple and they didn't work for sleeping because I side sleep as well. I thought the back position was a lot more comfortable than side sleeping on it. It took me many pillows to figure out what worked for me, and I still needed physical therapy to get rid of the neck pain. Unsure if headaches would be similar, but it could be. In any case, if the doctor really thinks a pillow could help, physical therapists do a lot more than exercise (mine helped with pillows and adjusting my home office chair, in addition to massage and exercise).