It certainly sounds a crystal is toggling between fundamental and 3rd overtone.
6MHz is possible to make as a 3rd overtone
The cylindrical crystals tend to be watch type 32kHz
It suggest that the 6MHz Xtal Oscillator is at fault. Since it would have the CMOS gate and caps internal, it should not be sensitive to load capacitance or orientation. 3rd overtones are usually thickness regulated to control and sometimes helped with an LC bypass.
So I suggest testing my assumptions;
- apply Finger tip capacitance on either crystal pads
- apply pressure with a pencil erasure tip. on either side of Xtal
- use your hand as an EMI shield touching board common and near xtal.
- gravity might apply to a microscopic disconnect inside but would be affected by minor stress.
My hunch is an electrical contact loss inside the unrepairable 6MHz oscillator from inductor to pad not vibration or EMI or capacitance will fix.