I find it interesting that the fridge would not work on electric but on LP when out of level. All the LP or electric do is provide a heat source for the absorption "boiler". To the best of my knowledge there is not a tilt mechanism installed in any OEM RV fridge. There is an aftermarket add on unit (and I have managed to misplace the link) that will measure the degrees out of level and monitor the boiler temps and shut things down to keep from damaging it.
As far as towing, from what I have read over the years, as well as in my experience, the basic consensus is that as long as your unit is nearly level while towing the occasional hills are not an issue, the vibration during the towing helps keep the chemicals from crystallizing, which is what leads to the system failure.
We have "burped" non working fridges and got them working again. You have to remove the fridge from the RV, then lay it on each side for a few hours/days, then when you get up right again, you hook it up and see if it will work. I would guess we have about a 70% success rate doing that. Doesn't hurt to try and might save you from having to purchase a new one or a new cooling unit.