Here are your parts
Parts for Kenmore Dryer 11064962300 - AppliancePartsPros.comIf you go to the Sears parts site they have actual part breakdown diagrams but they do not share these with other vendors.
I do not know where to get just the thermal cutoff. It would be best to talk to a AppliancePartsPros part representative.
You are correct that the kit you used is not the one that is recommended for this unit re: a 250 degree hi-limit and a 309 thermal cutoff.
Though I must also agree with the local person that it should work. Note that I do not have personal experience with this so it is a bit of a guess.
You may have another problem in the unit.
The correct kit is AP3094224. It has a 250 degree hi-limit and a 352 degree cutoff.
The hi-limit should have cycled so that the thermal cutoff did not blow.
The thermal cutoff is an additional safety device for the heater.
Note that the hi-limit's contacts are not as good as the main control thermostat on the blower so they do not last as long.
It could be that the main control thermostat is bad but it is more likely that the heating coil is grounded.
You can do a rough check of the operating thermostat by heating up it's metal face on a stove element. I use my soldering gun. Usually you can hear a click when it opens, then measure across it with a meter.
Just be careful to not burn your fingers.
To check the heating coil unplug the unit and both wires at the heating coil.
Measure across the coil. Should be in the 8 to 12 ohm range.
Then measure from each side of the coil to it's case, both should be infinite ohms (open). If not the coil is grounded and should be replaced. This can cause the coil to heat and the thermostat cannot regulate it.
I am off to the lake today so wuill not be available till next week.
If you do not own a meter, I would suggest you purchase a one. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.
4. When you start always short the meter leads together. This will tell you that the meter is working and if there is any 0 offset.
There is a good STICKY at the start of this forum about it's use.