Have you emptied the tank and tested the bladder? If water comes out of the air valve then it's probably bad. If no water comes out and the tank will completely empty it just might need air. I would at least try to see if this tank holds air.
The main factor on the equipment size is can you keep it running without cycling very much. I know you said it's a 10 GPM pump but what does the pump really pump? That will depend on the pumping level and outlet pressure. If by keeping all zones going you can keep the pump from cycling then you are OK. If the pump "beats" the sprinklers I would either consider a cycle reducing valve or a VFD system, depending upon your needs.