Might the OP's issue be a problem with the wand switches themselves or their controller?
I think we're probably looking at several distinct issues. The phone pairing problems etc sound different from the failure to shift he first reported.
I've had Bluetooth pairing issues, where the phone won't unlock the car, and variations. Going into Airplane mode on the phone and back again seems to solve that one. It IS very important to update the Tesla App frequently, especially after an over the air car update. I've also had annoyingly slow response with the phone unlocking the car.
Don't blame Tesla. The whole Bluetooth spec is wonky, and the range is unpredictable. I was trying to use my phone while waiting for my car to be fixed, and couldn't hear the other party. I finally realized it was because it was going to the car that was all the way in the back of the service hall. And there's a ton of interference now from all the technocrap we've spent so much money on. Also most phones are paired with too many devices. It might be a good idea to clear out pairings you don't normally use.
But I've also had the "can't shift" problem, that one clears after a few tries, like the switch is intermittent. Working the wand(s) vigorously without your foot on the brake to clean the contacts might be worth trying. A Tesla tech confirmed it was microswitches.
And I've had a very annoying variation, namely one out of 4 times it wouldn't go into AP on a double tap. Sometimes nothing, and sometimes it would go into just Cruise instead, barreling ahead without auto-steer, which almost got me into an accident once before I got obsessive about triple-checking.
The service center tech drove w/ me, confirmed it was real, ordered the part. A week later (yesterday) they took 2 hours to replace the wands and the whole steering column assembly. The tech said they have changed vendors on this part, as they have been seeing these kinds of problems.
Since then it seemed fixed, but at least once tonight I think it missed going into AP again (no chime, no message, just not "catching" the request). I'll keep checking.
In my case this was the issue, it might not be yours. My Invoice shows:
-- "Right wand seems to not consistently engage AP. Sometimes I think AP is on,"
-- "but it's only TACC. Switch/sensor issue?"
"Verified, replaced steering column control module.Test drove operation restored."
"Correction: Module - Steering Column Control"
"Parts Replaced or Added"
"Part Quantity"
"M3 STEER COLMN CNTRL MODULE KOSTAL 1"
"(1097662-00-L)"
"BOLT RC M16x1.5x26 PC88 PTP (1036655-00- 1"
"A)"
"Kostal 10400691752 (1028053-00-A) 1"
My model S is now 3 months old and recently on several occasions I have put the car in park while my wife left the car and then been unable to get the car out of park and into gear without having to exit the car and re enter and start from scratch. 
