I have a four pronged dryer and a three pronged wall socket. Can I change to a four pronged wall socket without the ground wire?

Ground is always required If dryer doesn't require neutral, you can use ground + two hots by switching the dryer to a 3-prong cord.

No, you cannot change to a 4-wire receptacle if you only have 3-wires and no ground. In your case you need to change the dryer cord to a 3-wire cord and plug. Lowes stocks them at all locations.