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Replacement parts for WHIRLPOOL LTE6234DQ1 | AppliancePartsPros.com

Take a look at Item 47 in Section 4.
The 309 degree thermal cut-off is actually a thermal fuse so once it blows it will not reset.
That is why it is sold with a new hi-limit thermostat,.
So if your cycling thermostat's contacts should weld themselves closed, the heater will cycle on the 250 degree hi-limit and not blowing the higher temp thermal fuse. You hopefully would notice that the unit is running too hot and check into it before the hi-limit contacts also weld themselves together and bl;ow the thermal fuse. At least that is the theory.
Note always check/clean your vent system when a thermal fuse blows.

It sounds like you may have a grounded element. The element has sagged or broken and is touching the case/frame so it runs unregulated. Often you can see this by inspecting the heater.
A better check is to use a meter.
Unplug the unit.
Remove both wires from the heater
Measure it, usually around 10 ohms
Then measure from each connector to the case, both should be infinite ohms.