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