@jdarnold remove the sensor wire and ground it, the temp gauge should move to HOT. That would mean your gauge at least works. If it does not you may have to replace your instrument cluster. If it does move than consider another different sender. Make sure that it is one for a gauge not a temp light. Also, make sure that your engine block has a strong ground, You could temporarily connect a jumper cable from the negative of the battery to the engine block. Let us know what you get. As for the engine light, have your vehicle scanned. Since we do not know where you are, in the US Autozone offers a free OBD scanning. Let us know what you find.
This from the repair manual:
Disconnect Sensor - Ensure temperature sensor connector is in good condition. Turn ignition off.
Disconnect coolant temperature sensor. Turn ignition off. If temperature gauge indicates cold, go to next step. If temperature gauge does not indicate cold, go to step 3).
2. Bypass Temperature Sensor - Connect a jumper wire between the 2 pins of temperature sensor harness connector. If temperature gauge indicates hot, replace coolant temperature sensor. If temperature gauge does not indicate hot, go to next step.
3. Check Red/White Wire For Short To Ground - Turn ignition off. Disconnect instrument cluster harness connector C237. Measure resistance between instrument cluster harness connector C237 pin No. 9 (Red/White wire) and ground. If resistance is greater than 10 k/ohms, repair/replace instrument cluster. If resistance is less than 10 k/ohms, repair short to ground in Red/White wire. Check gauge operation.
4. Check Red/White Wire & Pink/Orange Wire For Open - Turn ignition off. Disconnect instrument cluster harness connector C237. Measure resistance between C237 pin No. 9 (Red/White wire) and pin No. 5 (Pink/Orange wire). If resistance is less than 5 ohms, repair/replace instrument cluster. If resistance is greater than 5 ohms, repair open Red/White or Pink/Orange wire.