The one that has the wire coming from the motor and the one coming from the control panel both equal 240v as each one by itself is 120v.
If I read this correctly you measured 120 at each end of the element and get 120 volts, then add the two so you assume there is 240 across the element.
This in not a good way to do it as you have no way of knowing if you are measuring L1 at one end and L2 at the other.
You have to measure across the element and see 240 volts to be sure it is getting the correct voltage.
As an example.
Lets say that L2 is not there.
Using Neutral/ground as the reference side of the meter you will measure 120 volts at one side of the element (L1 voltage).
But if the element is good you will also read 120 at the other side of the element as the element conducts the voltage. This is also L1 voltage.
You probably know the below but just in case:
The voltage into the unit is actually two 120 volt supplies with a common Neutral.
They are 180 degrees out of phase.
So when one is at positive 120 volts, the other is at negative 120 volts.
So between the two you have 240 volts.
There is no way with a meter to know if you are measuring L1 or L2 voltage with a meter.
Also you say that the element has continuity.
Did you measure the actual resistance. It should be in the 8 to 12 ohm range.
Just measuring continuity is not a good way to go unless you know what your meter considers as continuity re: the beep the meter gives).
Lets say that the meter considers 100 ohms as continuity. and the actual heater resistance is 100 ohms.
Then this would work as the heater will heat up with 2.4 amps going through it (576 watts) but it will be very low heat re: a good element at 10 ohms will pull 24 amps (5760 watts).
But lets say that the meter considers 1000 ohms as continuity and the actual heater resistance is 1000 ohms then you still get the beep.
The element then pulls .24 of an amp ( 57.6 watts) and you probably will not feel any heat at all.
I hope the above is useful to you.