Welcome to the forum, glad you found us! Unfortunately you are starting with a tough question. Mushrooms in the lawn are very difficult to control. And some of the things that might help will not be easy for you if you are allergic to the spores (which would be my GUESS about the powdery stuff making you sick).
One easy thing (otherwise, except for the possible allergy issue) is just to keep raking them out of the lawn. The stems are fragile and break easily. This doesn't really get rid of them because most of the mushroom is underground, but it keeps them from maturing and setting more spores.
We had someone post here that putting lime (not the fruit, the stuff you use for liming your soil) on them helps eradicate them. Have not tried it myself.
They are not real susceptible to fungicidal treatment because of so much of it being protected underground. You can try making a concentrated tea of chamomile and cinnamon (both with fungicidal properties) and spraying them. Can't hurt and may help.
So the main thing is to work on keeping a very healthy lawn so that it can out compete the mushrooms and as much as you can avoid the conditions the mushrooms like... they like dark, moist, organic, superfertile. So don't fertilize your lawn, water it deeply but rarely and not at night. Keep your lawn raked and dethatched and closely mowed (so you are mowing the mushrooms down frequently). If your lawn is shaded, work on thinning branches or whatever you can do to let more sun on it...
Hope some of this helps a bit! Best wishes and let us know what you try and what the outcomes are!