When you disconnect the coolant temp sensor, it should default into turning the fan on. So if it didnt turn on, definitely could be the relay. Have you checked the plug itself to see if its getting power? Maybe just a bad fan?

For the revving up on its own, there are some hidden vacuum caps on the side of the intake manifold by the oil filter. Make sure those arent cracked.

 

When disconnected the gauge reads zero. When connected it reads mid range usually and way up there when idling for a while. Everything that reads that sensor should be getting a signal, even the audible check gauge alarm. So it should be something on the power to fan side or the fan itself.

 

Good golly Miss Molly! The relay/fuse diagram is written upside down! When you hold it able to read the words, that is upside down from what you are looking at in the box. Anyway, relay must be good because I swapped it with the start/run relay and it starts and runs. A 30 amp FUSE labelled "mini-relay" looks good too.

Will test direct power to fan, might need to get heavy enough wire for that not sure if I have it.

 

Back to my 1996 XJ, looks like my nearest O'Reilly has all the A/C tools available on their loaner program. Including the vacuum pump and cleaner container.