So glad you are managing with your sweet bundles of fur!
The HEPA air filter is the only way to go. I purchased an I.Q. Air filter Healthpro series. Works great. Tis one of the more expensive ones, but lasts many years [ 20 for me so far]
Dust and moulds are my main triggers, so the usual list of no rugs, no drapes, [soft folding blinds-easy to vacuum] vacuum with hepa filter [ have a Tri-star], damp dusting, and thorough cleaning/dusting. I also wear a mask when I leave my apartment, so to prevent picking up any bugs. In winter, the mask helps modify the cold air.
When planning an outing, I check the air quality monitoring reports; pollen and the PM2.5. I keep all vaccinations up to date: flu, co-vid, pnuemonia, RSV, etc.
This is a life long battle, and we must find the things which help us stay safe.
All the best in finding your safe environment.