It’s hard to know what the problem is without having the proper test equipment, like an oscilloscope, and knowing how to use it.
It would probably be easier and safer to use a 12V supply for the 12V strip(s).
You could also consider a 24V to 12V DC-DC switching power converter to get 12V from your 24V. Something like this:
Well, I did the circuit… I’ll just have to wait a few days to see if it fails 
With many loads where a resistor is set to limit the current flow very good place to measure the current is the resistor itself! Naturally you need to know the value of limiting resistor (usually the printed value is accurate enough) and measure not the current but voltage drop. Then calculate the current. I=U/R. That said, you need a DC power not the PWM, off course.