You can confirm a stuck contactor pretty easy.
Unplug that block again by the unit while it is running but the thermo is calling for nothing.
Take the cover off the business area of the unit, should have labels on that area, and a mess of screws to take it off.
Look for this, with some serious looking electrical wire coming/going from it, and some finer thermo looking wire going to it.
And you'll probably find it jammed closed. Just tap the unit with the end of the screwdriver gently and it should pop open.
You may find a bug got in there and fried, holding it closed.
You can clean that area out too to confirm easy movement. Should be able to push the upper (relative depending on installation) plate down easily.
After doing that, plug block back in. Should be off.
Go turn thermo to call for cool. Should be on.
Go turn thermo off, should turn off.
Unplug block again. Put cover back on.
If it got stuck again, you can decide now that you have looked at it, how comfortable you feel in replacing it, or if more cleaning would be in order.
Don't spray in water!