1) It may be possible to make this work, but it depends on your circuit. The current limitation is for a whole device, not a pin. Each pin has a much lower limit, and if your 676s are driving something else, that has to be added to the total pin current. It's a highly unusual configuration. If you need to turn the devices off (sleep won't work), use the control pin to switch a FET to power them, don't directly power them.

2) It's not just O.K., it's pretty much mandatory. A good design will have a single ground. Keep tracks short and ground tracks fat.

3) It's not clear what happens to this common track when the 676's are powering up - do the 676's make this pin an output briefly and mess up the signal? Resistors to give the connections a bit of elasticity are a good idea, but your powering is most likely the root cause of the problems. The whole area of initial states and power-up / reset causes designers to rip their hair out regularly.

I'd guess a combination of 1 & 3 is the root cause of your problem.