The more I think about this, I'm certain that there's something wrong with the laptop itself. Regardless of the ratings on the AC adapter, you are pulling far more current out of the adapter than its rated for, as evidenced by the melting plugs. The heat is building up at the plug end because the connector has higher resistance than the wire coming out of the AC adapter.
If the adapter was the wrong voltage, chances are the laptop wouldn't work when charged. If it couldn't put out enough current, the laptop would simply charge more slowly in most circumstances. But what you are experiencing is the laptop drawing more power than the adapter's plug can even handle, which is probably upwards of 8-10 amps for a typical connector.
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-Scott
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