More likely that you didn't wire the plug correctly
take it in to a repair shop for check
SOURCE: connecting 4-wire recectacle 14-30r dryer
doesn't matter both are 110v lines but if you want look up to top screw connections the wires coming to those screws are also colored as long as the red and black are to outside screws dryer will work ok
SOURCE: switching the cord on maytag neptune dryer
I have the revrse problem I moved and mow my 3 prong needs to be changed to a 4 prong how do I manage to keep it working and not eletrocute anyone with the extra cable
SOURCE: Changing from 3 prong to 4 prong cord
The white wire goes in the middle between the 2 hot wires. 
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SOURCE: AMANA dryer model ALE643RBW burnt wire
I would think that:
Low voltage. A lack of proper voltage increases the current flow (amps) and can cause overheating, thus, a burnt wire. Remember, the lower the voltage, the higher the amps.
Bad connection. Poor connections can cause small amounts of arcing, thus burning the connection. This adds resistance to the circuit which causes a possible voltage drop and, again overheating due to higher amps.
Since all that equipment is on the same circuit, I would really think it is a voltage drop situation. With everything calling for power at the same time, the breaker trips. In some cases, it may be just enough for the breaker not to trip but still have a low voltage situation.
There is the possibility of an over current, such as a power spike from the power company but it would have to happen more than just a time or two.
And the outside chance that lighting got into the system when it struck something close to you home but it, in almost every case, would destroy something else, usually electronic. (This has happened to me. Trashed my dryer, dishwasher, and answering machine.)
Good luck and hope this helps. Le me know how you come out.
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