Hey Dterrasi,
Do you have a volt meter?
If yes, with the oven unplugged from the wall, remove the cord. (Still the burned one I assume)

You'll have three terminals that the cord hooks up to. Left (hot), center (ground), and right (hot).
I'd check to see that you don't have a short from either of the side terminals to ground. So do a continuity (ohm) test by touching the two probes to an outside terminal & the center terminal (ground). You should have no change with the meter. I.e. No continuity.

Also, check the wall outlet that you are only receiving 110-120v.

The melting plugs indicates to me that it's drawing too much amperage. This could be a bad board or a stuck gas valve (?) but start with the simple free checks first. I would not trust this oven to be plugged in long until the issue is tracked down.