I clean with a steamer. It doesn't project particles, and it's extremely good at disinfecting. I also use a vacuum cleaner with exceptionally low dust re-emission. And an air purifier. The only "chemicals" I use are an extremely mild lactic acid based cleaner and a pre/probiotic on toilets and kitchen tops. And I don't use any fragrances anywhere. My allergy problems have disappeared. I really recommend this approach…
I am curious as well what steamer and products you are using?
I use a Miele C3 Silence vacuum cleaner. It's the cheapest (~€220) Miele vacuum cleaner that has top (A class in EU) filtration. More expensive models just buy you more accessories. Sadly, Mieles are super expensive in the US, and you can't do arbitrage due to different voltage. Nilfisk made in Denmark used to be extraordinary, but now they are pretty mediocre. /u/touchmyfuckingcoffee is a very famous vacuum technician in Reddit, and he recommends Miele and Riccar. He doesn't like bagless. Neither do i. Worse filtration and a mess to dispose if you are allergic.
I use a Coway 1008 DH air filter. It's cheap (€400), efficient, and uses low energy. IQAir has great reputation, but in some tests it has shown mediocre performance [1]. Perhaps these tests are not measuring particles in the smallest range, where IQAir claims to excel. I've seen some guys from IQAir at HN, perhaps they can clarify.
For house cleaning I use stuff from Sonett. Bio-D is also good, cheaper and easier to find. I also use probiotics from Chrisal. In the US, these might not be easy to find. I'm sure there are equivalent brands there. Check their ingredient lists. Bio-D cleaning stuff is so safe their hand soap is basically the same formula as their countertop cleaner. Many items are certified by Allergy UK.
I'm happy to hear any other recommendations from fellow allergics. I also found a two-stage filter for my shower (Sprite + Vitamin C Sonaki) did wonders to avoid spraying chlorine into my airways and reduce inflammation and itchiness.