Palmetto Bugs aren't like the typical German cockroach, and they don't really fly from place to place. They like dark and damp, and they can compress their body to get into some tiny places.
They mainly come out at night.
Indoor cats are good because they enjoy the sport of tracking and killing at night. They definitely keep the population down.
But the #1 best...and cheapest...way to get rid of them is simple: boric acid, tablets (my preference) or powder, placed in darp areas...like under and deep in cabinets, under the fridge, etc. Like 20 Mule Team Borax that Ronald Reagan promoted in the 50's. In fact, MOST anti-roach products have boric acid in them.
I'm from Tampa, home and Ground Zero of Palmetto Bugs, because we have so many...well....palmettos. They literally grow like weeds there. And the bugs follow.