Reverse Woes
I wouldnt' assume that.
If the computor is seeing the right current going to the Solenoid then it wont' throw a code.
It might still be bad.
thats the only thing in there I can see giving you that trouble.
With light side to side Movement you can feel a detent on the left , then it will center, and then you can feel a little detent to the right
pushing harder you will feel another one to the right.
With the car off can you feel the two detents to the right ?
My bet would be on that solenoid then.
You can try to remove it and see if that fixes your problem.

