High Idle/No shifting

Every so often (~once a month) my car will idle very high when I start it. It starts idling at around 2,000 rpms. If I try to drive it while it is idling high, it will not shift out of first gear. It will get to about 15 mph, then jerk back down to 12 mph, then repeat that over and over again. I let it run and that used to fix it after a while, but lately it has been taking longer and longer. Putting a dry gas solution in the gas tank also used to help, but I don't know if it does anymore. Today when it happened, the one thing that finally worked to get it idling and shifting normally again was shaking the car from side to side. I had to shake it 5 times and then the rpms finally dropped back to normal, and it drove without a problem.

In a seemingly unrelated note (but it could be related for all I know!), The car has been increasingly slow to respond when I press the gas pedal. It pauses and takes a second to respond most times when I come to a complete stop, which it never used to do.