Washers and dryers in public coin-op facilities are unknown factors, used by a myriad of strangers with a myriad of products that can leave strong scents in the machines, so is unlikely to be a fix for the OP's problem.

I would get one of those portable all in one washer and dryer machines. There are ventless ones that use water to control the dust. The towels don't come completely but they can be hung for a shorter period of time after being steam dried.

The OP says that there is no provision, no connections, for a washer & dryer in her apartment. The building has machines in a community/public laundry room, and dryer exhaust from machines in that public laundry room is inflitrating her apartment with scents/odors (and possibly lint residue) which is causing an asthma attacks.

You say you can’t move out right now. Is it because you signed a lease? You may be able to negotiate breaking it with a letter from your doctor. Are there units in the building with their own laundry? Can you at least identify a unit with greater spatial separation from the laundry room?

If you have room in your bathroom you could have a Miele W1/T1 dryer set stacked. The dryer runs off 120V and is ventless. The washer could be plumbed to the sink or tub. It will run off cold water only.

if you have smells coming in from other tentants or the on premesis laundry, then your landlord needs to remedy that issue. Vented laundry exhaust has to be located so many feet away from windows and air intakes to prevent just this problem.

if you truely cant break your lease (or its too expensive to do so) then try the options above.

The Miele models were designed just for your type of living space.