I see your post was over a month ago, so I don't know if you already corrected the problem or not... but Pick's string/wire idea is right on to help fish the new cord through. Just make sure you loop the new to the old, twist them together then tape them securely (with a nice, long taper) so they don't pull apart when you hit a snag.
My major concern (and reason why I'm replying so long after the fact) is why did the cord burn? The upstream breaker/fuse is there to protect it (and you) from that happening. Instead, your cord acted as the fuse. I would hate to think that the potential danger of this happening again is still there.
Please make sure the replacement cord is properly rated for 30 Amps at the purchased length. Longer lengths require larger cable (smaller AWG number) to help offset voltage drop caused by the resistance in the conductor. And please confirm your source of "shore" power is equipped with a 30A breaker/fuse (or less) to protect the new cord.
I would have to guess that the old cord insulation was worn, frayed, cracked, etc... causing the problem. Your trailer's main and individual fuses should have never allowed more than 30A to be drawn, but just wanted to give my $.02 anyway.
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Jason
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