Stop in for a cup of coffee

Hey Chris any ideas?
Hello. I need help. I have a 1963 fury with a 440 & assume a 727 torqueflight. when the car gets hot,(after 15 miles approx.) (not over heating hot) when I stop, the push button will not chg positions, it acts sticky like its not moving. if I shut the car off it will not start for about 30 min. I assume because the car is not in neutral & activating the safety switch. but after the 30 min the car will shift into neutral & start. I assume there is some sorta thermal expansion going on with the system? is it the cable? the shift mechanism in the dash? the safety switch in the trans? im at a loss here. thx for any help. Don.

1963 Fury push button issue
My guess is cable. That or he’s running the wrong filter and it’s blocking off fluid and causing it to run really warm