Not in a great place to help, I think you said a new one is on the way. sounds good.

1--/\/\/\/\/\/\/\---2 A resistor is usually drawn kinda like this to represent a long wire resistor. The wiper is then a contact that moves along the wire. lets say pin 3.
Measuring between 1 and 2 will be a constant resistance.
Measuring from 3 to either 1 or 2 will give a number that changes as you turn the wiper.

You neede to measure with the connector off - no power. don't need to measure with volts connected.

Once you say you got 4.5 on the 1K scale so that is 4.5k ohms. And possible. but nothing between 1 & 3 nothing between the others, maybe measurement error. You should get a reading between any two - all pairs.

one pair never changes other two change.

And you say you got this I think with connector on and power on? how are you changing the wiper position?
DC volt 10 Good
1&3=4.8v Probably good this looks like the resistor
2&3=3.2v Should change as wiper moves
1&2=.7to.8v rapid fluctuation Good!! Tells me the wiper is #2 and it is at the bottom of travel maybe fluctuating with idle air movement?

Trouble is the 2,3 voltage should be fluctuating too, up at the 3.2V. I would expect the .7 to go higher toward the 3.2 as you raise the rpm.

I'm sorry, I don't know what moves the wiper. If you take it off I'm sure you will see a hole for a shaft or a shaft that goes into the air horn box. Find a way to rotate the wiper while measuring the resistance from 2 to 1.

I would expect it to go from 0 to 4.5k ohms and 2 to 3 to do the opposite.
It needs to be a smooth change, no skipping or jumps. Some one has posted pictures of the thing appart and it shows the wiper and the circular resistor made of carbon deposited on a ceramic substrate. His wiper had worn through the carbon resistor in under the wiper so he just bent the arm of the wiper to move the wiper over to good resistor and it worked. (for a while not a ling term fix)

Gary