The play in the clutch cable is separate to the play in the shifter mechanism.
You will have to check and correct both.
Why do you say the clutch cable is broken? If it is it needs replacing, if not then adjusting.
This is a pretty common malady for the 4L60e transmission in GM trucks of this model. IIRC, the problem is most likely the sun shell is broken. If you were a mile away, I'd suggest you could drive it. Since you are talking 20 miles, the fastest you could drive it (reasonably) without causing issues for your engine would be about 20mph. That would mean it ...
No the transmission bracket do not break as a result of changing tire pressure.
It is more likely this is a result of not properly adjusting the engine mounts and transmission bracket when your clutch was replaced,or the engine mounts might have started to fail as a result of wear.
I think you should take a look at the engine mounts to see if the rubber is ...