The difference in corruption between government types (other than Communism) is fairly modest. Despotism -> Republic can be noticeable at times, but I'd never change governments for the corruption changes (except occasionally Monarchy-> Communism). Courthouses have a much larger effect on corruption. While non-Communist governments will have little effect on useless cities, courthouses can make them halfway decent. My guess is you'd get 4-6 or so production usable in New Brandenburg. Not great, but a better improvement than switching governments.
Specialist farms is the typical recommendation for such cities. I actually usually don't care enough about them for that, but if there's a hint of productivity (2/3 production instead of just 1), I might make them halfway useful with a courthouse. Communism makes corruption near-equal across your empire. Whether that's good depends. If you have few courthouses, switching to Communism will make your core less useful, not help your outer cities that much, and you'll likely see a decrease in both your production and your commerce, relative to Monarchy (and an even steeper decrease in commerce relative to Democracy/Republic). With sufficient courthouses and a large enough empire, Communism can be beneficial versus Monarchy in production and commerce, as well as military support. But it's very situational - Communism at its best means a lot of good cities instead of a fair number of great ones plus a lot of poor ones, with higher production than any other government, and probably higher commerce (than Monarchy, probably not Republic), but higher maintenance costs. Ironically, Smith's Trading Company can help a lot to offset the maintenance in Communism. At its worst, Communism means all of a sudden none of your cities can produce anything in a reasonable amount of time and your science funding is way lower because your cities with libraries and universities now have more corruption.
I'd be curious if anyone has more specific info on how much of an effect (percent-wise, for example) the various governments have on corruption. Specifically, what the difference between Rampant/Nuisance/Problematic/Minimal is.