The coating is quite thin, it's easy enough to use an oil instead, however the oil doesn't tend to stick to the existing Teflon so will be squeezed out.
Id clean the nut first, as there's probably something in that that's damaging the Teflon. Then possibly use grease as that stands more of a change to stay within the nut.
Although the best option is to replace the screw and nut. Ball screws are better than Teflon coated lead screws and often cheaper. Just get the same lead so there's no messing around with software. Just a bit of mechanical fiddling to fit them. A local machine shop should be able to accommodate you. Or.. You can often ask the supplier to make custom ends to match the gerber.. You've just got to measure all the dimensions correctly.