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Parts for Kenmore Dryer 11064962300 - AppliancePartsPros.com

See the attachment for the wiring diagram.

I am assuming that you are using the correct terminology and the cutoff you replaced is the one on the heater.
I am also assuming that you have had this unit for a while. The reason this is important is that sometimes purchased used units have had their wiring messed with.

Below is my standard blub when replacing this.
Check the heating coil.
Unplug the unit and both wires to the coil.
Check it with a meter, should be around 10 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 on high and the thermostats cannot regulate it so the thermal cut-off blows.

The hi-limit should have regulated the temperature so the fuse did not blow, that is why there is a new one with the thermal cut-off..
Note: That unless there is another problem in the unit the hi-limit should never have to open. It is just a safety device with the fuse being a backup safety device.

Just in case it is not a grounded element.
With all the below the high limit will also have to be replaced.
Check that the belt is OK.
Check the seals (drum etc) in the unit. The air is pulled over the heating coils, through the drum and pushed out the exhaust. So any large seal leak will pull in room air and the cycling thermostat on the blower will run the unit hot.
Check that the lint filter is not coated with fabric softener residue which greatly reduces air flow.
Check/clean your vent system.
Check/clean the blower wheel.

If all OK you may want to replace the cycling thermostat as it's contacts may not be opening (welded shut)

[COLOR="DarkGreen"]Looks like you have done most of the above so this may be a hard one to fix.[/COLOR]

You may want to try the following, just to be 100% sure the the vent is OK.
Unplug the unit and disconnect it from the vent system.
Now try a run.
If it now dries OK odds are the vent system needs checking/cleaning.
Be sure to check that the louvers on the outside of the house open correctly.

If you do not want any lint in the house.
Take a pair of pantyhose.
Put one leg into the other and then attach this to the dryer's vent.
Leave enough room so the pantyhose can balloon out like a windsock.

Both the above will let you check the temperature and the air flow.

[COLOR="Blue"]why would the thermal cut-off fuse EVER blow if the thermostat is functioning properly?[/COLOR]
It should not blow that is the problem.
Usually this is due to a grounded element. It grounds out is such a way that the element or part of it gets too hot and the high limit opens but this does not shut the element off since it is grounded. The cut-off then blows.
But you have checked this.

Now what happens if the element is not grounded and for some reason it is always staying on. Then the hi-limit regulates it and the thermal cut off does not blow. The hi-limit thermostat contacts are not like the cycling thermostat contacts. They are not made to constantly switch power on/off so after a while they fail, often welding together.
Then the thermal cut-off blows.

You also need to check inside the air duct on the back of the dryer that connects the heater housing to the drum inlet. Then you need to check inside the blower housing and the exhaust duct that goes from the blower housing to the back of the dryer. Make sure the blower wheel is tight on the motor shaft and not slipping.

Eric

Another update:

After venting the dryer into the laundry room for several days (with no problems), I reconnected the vent to the attic/roof line and all seems to be well. I've been able to successfully dry several loads of clothes, and they are drying MUCH quicker than they have been. Thank you all for your help. If this ever happens again, I have two fuse/thermostat pair remaining, and I have my homemade devices for cleaning the dryer as well as the vent system.