Synthetic motor oil.
Yeah, I know - no petroleum products shall be used, etc. But the synthetics are all high distillates so the incomplete burning problem is mitigated. A little goes a long, long way, however, so use it sparingly, like wet a rag with a thin film and rub it on with that. And it has the advantage of being widely available.
Also, any of the mineral oil (again, high distillate petroleum product) based rust preventers/lubricants like Ballistol will do the job. In this case, slather (love that word) it on. Try the fishing supplies section of the local hardware. Probably not a very big section in the Mojave, though...