We just replaced the heating element in our Kenmore dryer because it wasn't getting hot. But now the coils won't turn off, unless we completely unplug it from the wall. We have no clue what it could be.

Sounds like a short, which is probably what blew the first set of coils. This is a dangerous condition and you should probably call a repair technician. Whatever you do, don't try to use the machine until this has been fixed!

If you are determined to try to fix it yourself then--if it were me--I would unplug the machine, then remove the access panel to the electrical components (usually behind the top of the machine) and look for a place where two (or more) wires or contacts are touching when they shouldn't be.

This is commonly caused by worn or melted wire insulation that allows a wire to come in contact with another wire or some other metal surface.

It is impossible to be more specific without knowing the model of the machine and having some sort of diagram to look at.

If you have a multimeter, and a schematic diagram showing the machine's electrical connections, you can use the "continuity" function to try and trace the fault.

One thing to consider is that many house fires each year are caused by dryers and they often run off 220 VAC outlets instead of 110 VAC and so are a much greater fire/shock hazard than most appliances. So I will say again: unless you are very comfortable working around electricity I would call a repair person.

Hope this helps.

--John

I will agree with John as this problem can get complex and it may be best to call in a pro. Most dryers have 220 volt elements. That means that it must received two opposte 110 volt "legs" to me the element energize.

One leg goes through the timer, high limit, thermostat, etc, while the other goes through the centrifucal switch that is located in the motor, it is attached to the shaft of the motor rotor, and opens/closes contacts as the motor spins.

The fact that your dryer element is heating all the time, even when the dryer is not running, suggest to me that the centrifucal switch is staying closed.

One way to know for sure is to ohm out (do a resistance check) the centrifucal switch while the dryer is unplugged.

Perhaps a wire did short to the chassis of the dryer, but I would think, provided the dryer was properly grounded that this would trip a breaker.
Try ohming out the centrifucal switch on the motor.