So far it sounds like the contacts in the big relay that sends power to the compressor and fan motor are burned and have welded themselves shut.
If you are safe around high voltage power, turn the power off to the unit and remove the service panel on the unit that covers all of the electrical controls... follow the two very heavy wires that come from the disconnect switch on the wall outside of the unit... they will land directly on terminal 1 and terminal 2 of that contactor (marked T1 and T2), a close examination will show that the contacts are worn badly, and have welded together.
There is a 98% chance of that being the case, take a look, tell me what you find, attach a photo if like using the paper clip icon on right end of the tool bar above.. and I will coach you further and tell you how to order the new part ( $50 item or a bit less).
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