When attempting to start the car the starter will not disengage

Just so I understand the starter is not disengaging until you remove the battery cable?

If that is the case then we need to look at voltage coming from the ignition switch and then to the solenoid. It could just be a starter issue but clearly we need to find what is still supplying the voltage to the starter.

With this being altered, I am not going to be able to provide much supporting documents but a Chevrolet 350 is pretty straight forward.

Let's tackle the starter not disengaging first then we can address the no start.

Okay. More then likely the negative battery cable melting is most likely a different issue. That is most likely a poor connection or a worn cable. Basically the high resistance that this creates causes excess heat and then it melts.

As for the starter not disengaging, we need to track the wiring from the starter to find out where we have voltage on the control wire. You should have 12 volts on the wire from the battery but the control wire that works its way back to the ignition switch is the issue. You are getting 12 volts on this wire when you shouldn't. It should cut the 12 volts when you take the key off the start position.

Please keep me posted. Let me know if you have questions. If so, maybe get some pictures of your engine bay and components of the starting system so we can see how yours is set up. I imagine it is pretty standard for a 350 but seeing the components may help as well.

Let me know how you make out. Thanks