I don't have hyperacusis or tinnitus, but I am extremely sensitive to noise, especially fans, engines, pumps, and anything else that makes percussive or resonant noise. I am fortunate to live quite remotely in the countryside, with no human sounds nearby. I have a personal policy that nothing that makes continuous noise is allowed in my house. I installed all my appliances in another structure (a root cellar) a short walk from my house. It's made all the difference!
If you live in the typical American configuration of a house with a garage that is near the kitchen, consider putting your fridge/freezer out there. Sure, it's a few steps of inconvenience, but you'll find the lack of noisy appliances a true consolation. If you really need quick access to something refrigerated, put a cooler in a corner of the kitchen, and make ice with the freezer that's in the garage.
Truly silent refrigerators exist, but you won't find them at your typical home improvement box store. (Ask me why I know this!) Look into Amish appliance suppliers. Or the tiny-house folks who've found quiet, compact absorption-based fridges.