Well, I've tried several things.
I couldn't find any milk glass or thin diffusing "filter", so I used tissue, which works great, although if you're not careful, you get texture. It also helped with highlights when enlarging. I'll look for a plain white filter similar to Ilford's MG filters.
I contact printed without the frame. Just negative on paper, with glass on top. This helped a large amount. Apparently, the backing of the frame has more give than I thought. It wasn't noticeable before, which brings back the condensing light.
I also washed the glass with alcohol and a lens cloth, letting it air dry. This also improved things.
Doing each individually was an improvement, but putting the 3 together seems to be the ticket. On only 1 print did I get what may be something bad, but it's hard to tell if it's a fingerprint, or something else.
I also considered sbuczkowski's comment about previous contacts. I reviewed several at random, including earlier attempts with the same negatives that showed rings. Using a 10 loupe, not a single one was visible. Perhaps it was luck, grace, or something else that prevented them appearing.
When I get more paper and time, I'll add drying the negatives with a hair dryer, and possibly washing them with alcohol, and see what that gives me. The only paper I have left is an unopened pack of 25 sheets, 20ish year old Kodak Polycontrast 5x7. Even being unopened and stored in a cool, dry, and dark place, I think it will be unusable.
Thank you all for taking the time to reply, and for sharing your hard earned tips and tricks! I really appreciate it.