2013 Fusion running hot

It sounds like you have 2 problems at first blush: the car's overheating light comes on even when the car's been sitting, and the car won't roll even though it's started. However, if I remember, newer Fords have a computerized gear selection system rather than using mechanical linkages, so the 2 could be related if there's a computer fault. Or it could be the computer isn't putting the car into gear because it has detected an error condition.

If a computer failure is causing both problems it's an expert job to fix it, or to replace it, if it's separate problems then each has a cause that can be independently solved. Taking the overheating problem first, if the light comes on even when the car is cool then it's most likely a problem with one of the temperature sensors. That's good news if so, sensors are generally cheap to buy and easy to replace. I've been told there's a transmission solenoid that frequently causes issue with cars not going into gear, also a cheap and easy fix.

The best thing to do is get your cars On-Board Diagnostics (OBD2) codes read to find out what the computer is saying is wrong. If you can get your car moving then take to an auto parts store, many of them will do this for free. If it won't move you may have to pay for someone to come although you can buy a bluetooth reader and read the codes yourself for less than $20.

It’s the engine coolant temperature sensor. When it goes bad it automatically registers as overheating. When that happens, the engine goes into failsafe mode which is why you can’t accelerate.