Hmmm this is a strange one. This is a guess but it sounds like either the ignition system begins failing after 10 minutes or the EFI computer is having a problem. Intermittent problems are always the hardest to troubleshoot. Is the check engine or service engine soon light coming on? It could also be a cam or crank position sensor that is failing when it gets hot. The best way to determine this issue would be to take it to a GM dealer and let them look it over. Their mechanics know this car, have been working on them for some time, and you can bet they have already seen this before and will know exactly what is wrong. It sounds like it could be dangerous so I would have it looked at as soon as you can.

You can't wire it, if the shifter is on the console between the seats, you probably will have to take it to a garage and have the entire shifter changed. Sometimes dirt builds up in the shifter lever and doesn't let the metal piece on the shifter make contact with the switch in the console. Either way it needs to go to a garage.