You and your cat are breathing in microscopic bits of roach shit and their decayed bodies. That's what triggers asthma for a lot of people. I don't know how much worse it is than the other shit that floats around everyone's dwellings but it's not pleasant to think about.

I have had the unfortunate experience of being building manager in a mixed-use building where someone moved in from a roach infested apartment, and brought them with her. They spread throughout the building within a few months and are presumably still there, though under control. If you move and bring roaches with you, you can be held liable for the fumigation costs which could be thousands. It would have to go to court, so you might not have to pay that much, but I don't think you want to be carrying roaches around with your stuff the rest of your life.

From my experience the only thing that really helped knock down the roaches was fumigating with chemical fog - we tried more eco-friendly stuff first and it just wasn't doing it. You have to put all your food and dishes away so they don't get fogged, and then leave the place for at least 4 hours. Supposedly the stuff they use is only toxic to animals with exoskeletons, but you still want to keep the residue off dishes and wash your counters down well, etc. After that it's usually monthly followup visits with liquid sprays occasionally, gel bait and indicator traps so the pest control guys can see where the problems are.

When/if you move you need to be really, really meticulous about sorting through your stuff and then moving it to a roach-free place right away. Don't leave cardboard in your house ever; they will crawl inside cardboard and they can move with you that way. Also don't leave stacks of paper around, cause they like to be between sheets of paper for some reason - I have seen this first hand. Make sure you familiarize yourself with the appearance of their egg sacks because they stick them in the weirdest places (like inside a microwave). They especially like refrigerators and other electrical appliances; when we were cleaning one kitchen we didn't see any roaches until we moved the fridge and the microwave, and they were spilling out of both. Ugh ugh ugh.

Also: roach poop looks like coffee grounds but slightly sticky. If you see that in your kitchen you might have more of a problem than you think. Pull your fridge out and check that.