2008 MKZ Coolant Temp Sensor

I’m new to the forum and I’m at wits end. I have searched and tried everything so I figured I’d try to see if anyone here could help. I have a 2008 MKZ. One day while coming to a stop the check engine light came on and the car started shaking real bad and wouldn’t go over 30mph. If you stopped and turned it off and then started it back up it would run fine but the light was still on. I put the code reader on it and it said the thermostat was the issue, so I replaced the thermostat. The next day it did it again. I know it has a good thermostat now though. The temp gauge goes crazy too. It’ll be normal then it’ll drop and act like the engine isn’t even warming up and then the gauge will peg out to hot and it’ll start shaking real bad. When the gauge stays on cold the a/c quits working. Then the gauge will go to normal temp and a/c starts working again. The cooling fan runs constantly too. From my research I believe it to be the coolant temp sensor going bad, but I can’t find the sensor. It’s not in the places that I’ve seen from Google or youtube searches. Does anyone have any idea if it could be anything else and if it is the temp sensor where is it at? Sorry for such a long post