I think you have an even bigger problem than the coax.
You may have a SHORT somewhere and you have a voltage that should never be on that coax to start with.
The only time there could be a voltage on coax is when feeding an external antenna amp and even that should be current limited which would not have enough current to melt coax insulation.
As far as replacing the coax goes, seriously doubt you will ever be able to pull out the original. It is most likely stapled into place and or may make a few very tight turns in the wall cavities making pulling impossible.
Best bet would be to bypass the offending coax by running a new run.
You will have to get creative in order to accomplish this.
If wall outlet is above say a table or cabinet, go below the surface of the table or cabinet then drill a small hole to fish the wire up the wall to the outlet..
Then run the wire any way you can find to the shore connection, through wall cabinets or even under the floor then back up to get it where you need to go.