Agree with the above^^. I'd be more inclined to think it is a loss of refrigerant rather than the reversing valve. The fact the a/c neither cools nor heats points to lack of refrigerant.
A stuck reversing valve will result in the a/c remaining in whichever mode it was in when the valve failed. Your system neither heats nor cools, which is a function of refrigerants flowing through the system, and that is not happening, apparently.
You are considering replacing the reversing valve "coil," which I assume means the solenoid. That is not difficult, but your symptoms are not consistent with a stuck reversing valve.
BTW, a 17 year old a/c on the roof of an RV has generally served its time.
Larry, Eileen, and Finley
2004 Alpine 36FDDS
Third motor home, first Alpine, no need for another.