You'll need some safety pins for back-probing the connectors, the way the ESM tells you to check it is incomplete. They also don't mention anything about pin-fit, and that's something you can also check with an 0.025" diameter safety pin. You may end up needing that since the sensor is probably falsing and not dead, but start with any old pins and we'll see if you need a specific one.
1) Check voltage between G/Y and BR with the sensor disconnected and the key on. You should see right around 5V. If not, check from G/Y to the battery ground. If you get 5V from the battery ground, but 0V when using the the BR wire, then BR is broken. If both grounds give you a 0V reading then G/Y is broken.
2) Check voltage on R with the sensor disconnected. You should also see 5V there, but the voltage is coming through a pullup resistor in the ECM and not from the 5V supply (AVCC). If you see 0V then either R is broken or there's a problem in the ECM.
3) If steps 1~2 check out, back-probe G/Y and BR with the sensor plugged in but removed from the valve cover. Repeat step 1 but using the back-probes, keeping the sensor away from metal. If G/Y reads much lower then you have resistance in the G/Y wire. If BR reads much above 0V then you have resistance in the BR wire. If R reads lower then it's a bad sensor, new or not. Resistance on G/Y or BR can be in the wire or at the ECM connector pins, so that's where pin-fits come in.
See what you find, and if 1~3 don't reveal anything, I'll show you how to remove the backshell from the ECM connector so you can check there.