Yup, that was it. A part of the coil on the heating element had sagged or warped and was touching the frame. I was able to pull it off without breaking it. I also had to replace the thermostat and the fuse to get it working again.

Thanks again for the advice.

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Replacement parts for Whirlpool WED5300VW0 29" ELECTRIC DRYER | AppliancePartsPros.com

See the attachment for the wiring diagram
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I read somewhere that it's probably the control board (a bad thermostat wouldn't make sense, since the heating element shouldn't be coming on AT ALL on Air Only). Does that seem right?[/COLOR]
First part is wrong, the control board just advances the timer does not effect the heat.
Second part is correct, because timer contacts BK/R should be open during air only disconnecting power from the heater circuit.
There is a possibility that this set of contacts are welded together.
Unplug the unit ans set it to air only.
Remove either the black or red wire (this prevents you from reading an alternate/parallel circuit path)
Measure across the contacts.
Should be infinite ohms (open).
I sort of doubt this is the problem because even if they are welded shut the unit should still regulate the heat output.

[COLOR="Blue"] Is there only one control board for this kind of dryer? I saw a part called 'Dryness Control Board', and I'm wondering if that's what I want.[/COLOR]
No, it only advances the timer when in an auto mode.
It monitors the moisture sensor. This is 2 parallel metal strips, when there is moisture between the strips a small current passes between then. The board senses this and holds the timer off. When the clothes are dry, there is no moisture so no current and the board then advances the timer to shut down.

Most likely cause is grounded element.
Check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 8 to 12 ohms.
Then check from each side of the coil to the case/frame, both should be infinite ohms (open). If not the coil may have sagged or broken and is touching the case. This can cause it to run and the thermostats and timer cannot regulate it.