Tach hookup

It does sound like an internal problem with the tach.
Just so you know, the white wire off the coil IS one of the coil grounds.

If I were in your position, I might try connecting it to the black wire on the coil that goes to the module.
It may interfere with the module signal, but I might try it just by skinning a small spot in the insulation and touch the green tach wire to it while it was running.