Here are your parts
Replacement parts for WHIRLPOOL LEQ8000HQ0 | AppliancePartsPros.com

Here is the wiring diagram
https://www.servicematters.com/d...0Sheet%20-%203977704.pdf

I assume the unit does start rotating and you are only missing heat.
See Item 9 in Section 3 of the parts

I would suggest you purchase a meter. 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. A little late for this but better late than never.
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.

There is a good STICKY at the start of this forum about it's use.

You could unplug the unit, remove the wires from the thermal cutoff (fuse), short them together, tape them up so they cannot short to the frame, plug the unit in and give it a try.

If it is blown replace it ASAP as it is a safety device. Replace both as shown in Item 9.

My next question is, can you run the dryer for a period of time by bypassing the thermal cutoff or will it shorten the life of the heating element? Do I need to replace the thermal cutoff immediately or can we dry a few loads of clothes without it?
You could do a few loads, problem is often a few loads turns into many loads.
I would also keep a close eye on it when doing this.
It should not effect the heating element but now that I have said that; it will (Murphy's Law)
So I do not recommend this as I would feel very guilty if something should happen.

Also clean and check your vent system and the blower wheel.
Lint filter, especially if you use fabric softener which can coat the filter screen and greatly reduce air flow. Would need to be cleaned with soap and water.
Also feel around the back and front of the drum to see if the seals are OK.
Often it is an air flow problem which causes the fuse to blow.