Okay, I was wondering about possible resonance between the fans.
Overall, I would focus on the fan versus a new case per se.
You mentioned that unscrewing the fan bit reduced the noise.
Use a rolled up sheet of paper or long cardboard tube as a listening tool to pinpoint the sound's location. Put one end to your ear and carefully move the other end around the fan(s). Then follow-up with a close inspection using a bright flashlight and magnifying glass. Look for anything that can vibrate, touch, etc. Cracks, chips, burrs....
Remember that as the fan comes up to speed all sorts of forces come into play and the fan casing, blades, etc. may all twist, stretch, flex, whatever, and something starts touching. Hence the buzz.
Are there any depressions as mentioned by
@Schlachtwolf
Cut and fit in masking tape or electrical tape as needed.
Just seems to me that doing some additional experimentation, even if trial and error so to speak, may prove to be worthwhile before going to a new case.
Are you able to take and post a couple of photographs? Someone may spot something amiss or otherwise of interest.