Bad receptacle and cord...
Chris; I realize I'm repeating what has been said, but I want to help reinforce the correct advice and dilute the not-so-good advice.
There is nothing wrong with your dryer which caused this. If anything is indeed wrong with the dryer, it's unrelated to the cord and plug failure.
When the dryer is heating, it draws approximately 20 to 30 amps. The motor is under little load so it will tolerate some voltage sag, therefore you won't see any significant change in operation of the dryer when there is voltage drop in the circuit.
I can see that your receptacle is old, and has been painted. Over time, the contacts in the receptacle lose their spring tension and no longer properly grip the terminals of the plug. The copper oxidizes; building a layer further adding resistance and hampering a good electrical contact. Add some stray paint onto this, and you have a very poor connection with high resistance. Heat is generated when current passes through a resistance. Inside the plug and receptacle, there isn't much of a path for heat to escape, so it builds up even if the voltage drop is seemingly small.
Once the contacts in the receptacle reach a certain temperature, the metal is annealed and softened, and the contacts never will grip the plug properly again; even if they are cleaned and bent back into the correct position. Receptacles don't typically show external melting, so you can't tell the condition of the contacts by looking at the outside of it. There is a chance that numerous previous dryer cords have failed in this receptacle and it is in far worse shape than the appearance would let on.
The only way to prevent continued failures is to replace the dryer cord and the receptacle at the same time. If you aren't confident to do this yourself, it shouldn't cost too much to hire an electrician for a simple repair like this.
Sincerely,
David