Distorted waveform is because of the harmonics, which may be because the resonator is overdriven. The datasheet you posted says the drive level should be between 10 and 100 uW. You measured 1.2V of the drive voltage. The datasheet also says a 20MHz resonator equivalent resistance is around 50 Ohm. That gives 1.2^2/50= 28.8mW which is two orders of magnitude higher than recommended drive level. Try inserting say a 10 Ohm resistor in series and measure voltage across it, calculate the current and then calculate the drive level. I am not sure how to reduce the drive level, perhaps a attenuator may need to be consructed. The 500 Ohm resistor seems to be in the kit to serve that purpose, but i'd think a series resistor will reduce the resonator Q. You still can try it. A better way is perhaps to replace the resonator with one with appropriate drive level. But as other said the micro will work as it is. There may be technically a risk of damaging the resonator with time if it is overdriven too much.
If you will be measuring RF voltage across the series resistor with an oscilloscope make sure your circuit ground is floating, i.e. not connected to the scope ground, or use a 2-channel oscilloscope with math to emulate a differential probe.