There's a lot of noise in the distortion. Yours does sound more noisy than typical. It can get very noisy if you have the Amount control and the Tone controls all the way up. When the tone control is at the maximum I have left a lot of high frequency content so the high frequency is not filtered much. Also, there is a lot of gain in the circuit. Try turning the Amount control and Overdrive control to 50% position and see if it's much better, this is equivilent to the max settings on typical distortion units.
Other than the noise the distortion sounds correct. It's a very compressed sound with a bit of a hole in the midrange frequencies. The overdrive is more gentle and more even frequency response.
About testing the voltages with DC or AC setting you should be doing all voltage measurements with your multimeter in DC mode. The only signal that you need to test that are AC is the actual audio signals, and this should really not be done with a meter but instead by using an oscilloscope or listening to the signal. I know you don't have an oscilloscope so you can actually make a cable to plug into your mixer and listen to the signal at various connections.
Do these tests first:
1. Put your multimeter in DC Volts mode.
2. Check again IC301 pin 8. it should be 12V
3. Check IC303 pin 3. It should be 6V.
4. Check SW303 pin 2. It should be 6V.
If this is OK then do this...
1. Get an extra cable and cut off the end so that you can plug one end into your mixer and use the other end to probe the circuit.

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2. Connect a capacitor to the center conductor of the cable. Any capacitor will do, 1uF/10uF/100uF the value is not important. Connect the (-) side to the cable. You can use another type of capacitor too if that's all you have... tantalum or polyester type is ok. This is just to eliminate any DC bias from the signal. If you don't have an extra capacitor it is ok not to use it.
Next, connect a piece of wire to the shield...

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3. Connect the wire from the shield to an extra pad C on the x0xi0 PCB.

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4. Now you can use the other end of the capacitor to probe and listen to the audio signal.

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Now you can plug the other end of this cable into your mixer so you can actually hear the signal as you probe around.
1. Listen to signal at pad B. You should have signal here because the Distortion works. This point is before the overdrive/distortion circuit.
2. Select Overdrive with 'Type' switch.
3. Probe the signal at IC310 pin 1. You should hear the distorted signal at high volume.
4. If you hear good signal in step 3 then probe again and listen for signal at IC301 pin 7. you should still hear distorted signal at good high volume.
5. If you hear good signal in step 4 then probe the signal at pad A. This is the output of the overdrive and you should hear the signal here too.
Let me know how this works out.