That sounds like the Resettable Protector may be bad.
Check the voltage across the Protector when the pump should be / is running, to make sure it is 0 VAC. (Closed contact)
It sounds like you have some resistance in the circuit, somewhere.
Seeing that Hot Melted connection, and just letting it go for months, Was a Big Error on your part. Sorry, I speak the Truth Most of the time. lol
Be careful playing with electricity.