You probably not only had Newton's rings in the old setup as well but, since you say the light had a large diffuse component coming from other angles around the room, you probably had multiple instances of them with different radii. This might have smeared them out so they were less visible than they are now.
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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.