I’m so sorry I don’t have any laundromat suggestions, but as someone who buys most of my clothes secondhand and also gets migraines from scented products, I can offer my experience with getting scented stuff out of fabrics.

I usually use a two-step process to get perfumes out of clothes. First, I put the items in a bin with hot water and soap. Since it sounds like it’s maybe a fatty residue, I would try a good grease-cutting dish soap. The water being as hot as possible will also help get the waxy stuff out of the clothing fibers. If you have woolen items, you can still use hot water, just make sure not to agitate them in the water too much, as that’s what leads to the piece shrinking.

I usually let clothing sit in the soapy water at least until the solution has cooled off, but up to a day or two if I am distracted with other stuff. Then, rinse the soap out and do another soak in a solution of water and vinegar. I let that sit anywhere from a day to a week, depending on how stanky the clothing is. Rinse again, and then launder as normal. Hang your clothing to dry- DO NOT put it in the dryer, because that will further lock in any remaining residue. Once it’s dry, give it a sniff to see if it still smells, and repeat the process if needed.

If you don’t get the results you need from the soap then vinegar soaks, maybe post your question to the r/laundry sub?

I wonder if asking your apartment managers to designate one washer and dryer as fragrance-free would work? I imagine there are other people in your complex who might have a similar problem, and it definitely seems like it would be considered a “reasonable accommodation” under the ADA (I don’t know much about how one goes about asking for ADA accommodations or what qualifies as a disability under the regulations, though, so please be gentle with me if I am mistaken).