'If the fan is coming on when the AC is turned on, then it is most likely the fan resistor.'
Incorrect. The fans both start out on low with a/c on and both ramp up to high speed at 250+ (or 275, I forget) psi a/c high side pressure. In other words fans don't work on a/c either if resistor is bad except under certain high pressure conditions. Fans are on all the time a/c is on but can be on high or low. I've watched my '02 ramp up and down in fan speed while charging the a/c.
Resistor checking is not rocket science, it is bad if the connectors are melted and the thermal fuse can be simply checked with a VOM, if open like a normal fuse it's bad. It should have very low resistance.
Mine burned the thermal fuse making it bad as well as melting the entire connector. I gutted the plastic to drill contacts for a positive screwed on connector, with new 10 gauge wire I reused it to save the price of the part. Works fine. Seems kinda stupid to put a thermal fuse in there if it can still melt bad enough to possibly cause a fire. Found it by the wisp of smoke coming from it.........
Extremely rare for a piece of resistor wire to simply go bad, you can recover the part with a bit of thinking. Easiest $60 I ever made.
To OP, check the resistor for continuity, if good go for low speed relay next, it can be bypassed to check both the circuit and the relay itself. Have you checked the fuses for it yet, shoulda done that first.
Fan power flow is through fuse (for high and low speed), then relay, then another lower rated fuse I'm thinking for low speed only, then resistor if low speed. Sounds like you've lost the low speed circuit. Also check both fans and see if they spin freely by hand, seized up will spit fuses.