Yeah, corruption's tough, isn't it? You said you switched back to republic from democracy, was that due to war weariness? Because corruption is actually lowest in democracy.
At any rate, here's what I do with ultracorrupt cities (those producing only 1 uncorrupted shield per turn). First, I sell off any unnecessary improvements. For example, say I've captured a size 5 city that has a barracks, granary, harbor, marketplace, courthouse and aqueduct. I see that even with the courthouse it's only making 1spt. So the courthouse gets sold. The barracks gets sold unless the city's in a vulnerable position - this town's never going to produce any units for me. The marketplace stays if I control a lot of luxuries, since it'll let me keep a lot of citizens happy in the town for only 1 spt upkeep. The harbor goes unless it's only one (or one of only a couple) on the landmass, or unless I have Smith's to remove maintenence on it. The granary goes, probably (see below). I will probably have to buy a temple or library to expand the cultural borders, but I can sell it again as soon as that happens. The aqueduct I cannot sell; the game doesn't allow it.
I then bring in my workers to irrigate (and railroad if possible) all high-food producing tiles in the expanded city radius. Once that's done, I join workers of my own nationality or of an extinct civ's nationality to the town to bring it up to its maximum happy size (you might be able to go a little higher than that if you're done fighting and war weariness won't be a further issue). If I don't have any such workers to spare, I let the town grow itself to that size by working the high-food tiles (this is where I would temporarily keep a granary). At that point, remove as many workers as you can from the lowest-food-producing tiles and convert them to taxmen - the output of taxment is not subject to corruption. Generally for a size 12 city you can manage to support about 6 taxmen on decent land after railroads, occasionally as many as 8 or 9 on very good land. Finally, set the city to product wealth; that'll convert that 1 spt to 1 gpt.
The benefit: you've gone from a city that was costing your civilization 6-ish gold per turn to one that is producing 6-ish gpt. Multiply that over 30 or 50 ultracorrupt cities and it makes an immense change to your income.
One last thing: the lux slider at 30% seems a bit high. Unless that's due to war weariness (in which case it's fairly normal), you may want to consider building more marketplaces and cathedrals in your core (i.e. noncorrupt) cities so you can reduce that a notch or two.
Hope that helps,
Renata