20 minutes after getting gas, the check engine, vsc and trac lights all came on.

Clearing the codes manually never solves the problem that cause the illumination of the light. You are on the right track - remove and replace the gas cap, making sure that it is seated properly and tightly (at least one turn while 'clicking'). It may take several drive cycles for the light to go out on it's own. A 'drive cycle' is a a term to allow your ECM a 'relearning' opportunity. It is complete after certain driving conditions are met - different speeds, idle times, ignition on/off instances, etc. After 2-3 days, in most cases, you should have met the requirements. IF after 4 or 5 days the light remains on, purchase a new gas cap, install and 'repeat'. Gas caps ARE expendable items, contrary to popular thought. They DO go bad and the WILL need to be replaced throughout the useful life of an automobile.

Depends how much driving you do and how often. Just go get the codes cleared at a shop or parts store and see if they come back.