melting plug

If this is happening at the same park the PED needs a new outlet.

Are you using the same outlet all the time when this occurred or were you at different locations?

Is it just one prong of the plug and is it always the same prong?

Not suggesting user error, but are you hot plugging or removing a lot (put plug in or out with pedestal power switch on and heavy loads on); or is your plug shape preventing you from fully inserting it in the socket? Somehow you are getting more resistance (heat) at your contact.

I'm wondering if a 30 amp adapter would work for you. If it is a post problem the adapter might take the hit leaving the cable in tact. Just my $.03.

Are you replacing just the end or getting a new cord with premoulded end?
In general I think the replacement ends are better.

I recommend this one:
https://www.progressiveindustries.net/tt-30p

Although poor connection in the worn and corroded outlet is often the issue.

A 30 amp service can only supply 30 amps. If the current rises much above 30 amps the breaker will trip. If the equipment is properly constructed and in good condition it will not over heat at 30 amps or some design amount above that. Low voltage cannot make the current increase more than that 30 amps without causing the breaker to trip. And the RV cannot overload the circuit without also tripping the breaker. Again, if properly constructed and in good condition.

Heat is caused by too much current flowing thru a resistance. A bad, worn, or loose receptacle will cause the plug to overheat and possibly melt the plug. Likewise a bad, worn, or loose plug will cause the plug to overheat. A loose wire in either will do the same. All of those add resistance to the circuit.

You can burn up a 30 amp plug even on a 15 amp shore power supply. It has to do with low voltage. Then items such as the roof air conditioner draw more amps. That causes the plugs to heat.

I now limit my 30 amp shore power to 24 amps. I've not burned up a single plug in the last 12 months.

It is being overloaded for some reason.
I would start checking the connections in your rig.
Check all the connections in the 120 volt power distribution panel. Look for melted insulation.
Find someone with an Amprobe to check amperage of the rig when connected. It is possible that something is going to ground in shore power cord. Maybe screw through it or bad metal clamp.
Add you rig to signature, make and model. This would help identifing known issues with that particular unit from others experience.

Sounds like the park needs to replace or repair the socket in their pedestal. Poor connections cause heat.