Replace the face mount hinges with hidden ones

Just to stop the ugliness, I would just get a can of Rust-Oleum gloss enamel (black, white, whatever), then remove the hinges from the cabinets.

Clean them thoroughly, and if there's any existing lacquer or hardcoat, try roughing them up with a bit of steel wool. If you can completely disassemble each hinge, that would be ideal; otherwise, work a small amount of silicone lube or grease into the action, to help keep the paint out of the hinge itself (keep the outer surfaces clean, though).

Then, simply give them a couple good coats of the enamel spray; each coat should cover well, but don't go too thick or you'll get runs, drips, etc, as well as having paint over the tops of the moving parts (which will then crack the first time you move the hinge). Let dry, flip and repeat. When both sides are done, reassemble and reinstall the hinges. The enamel should cure into a nice hard shell that should last you until you can spend the money to replace the hardware, or redo the whole kitchen.

If you are particularly interested in keeping the same door hinge system then it may become necessary to take down the cabinet sections where the hinges need to be removed. You already know that the cabinets were installed in a modular manner after they were built due the pivot hinge mounting. Taking down the cabinet would involve first emptying them out. Then search for screws inside the cabinet that go unto the wall studs behind the cabinet or screws that go up into the ceiling or cabinet soffit. In some cases there may be wood strip trim pieces along the cabinet edges against the wall or ceiling that need removing.