S really simple, no tools way to check if the contactor is stuck closed, would be to turn off power to the JO box, and turn on power to the pump contactor. If the pump comes on, then the contactor is bad, and you can rule out anything in the JO box. The pump contactor is NOT in the JO box. It should be it's own box, mounted on the wall. You can "trace" the 1" black flexible conduit from the motor to the wall. The contacotor should either be siemens brand or Telemecanique.
You can replace the contactor with any brand, as long as it matches up HP wise, and has a 24vac coil. If you change brands of contactor, you will need to buy the same brand of overload. The only exception, would be if you have Telemecanique, you can use schneider electric brand as they are the same.