You should have all the ground wires connected and that should not cause any interaction. Try disconnecting the downstream outlet wiring from the second GFCI for diagnosis. If there is a neutral to ground short downstream then one or the other GFCI will trip. If this is the cause then the two GFCI receptacles will then work ok and you can leave them alone while looking elsewhere for the real problem. If that doesn't solve it, then recheck carefully that every wire is on the screw you intended and nothing is criss-crossed. For instance if you ran the incoming hot and neutral to different GFCIs and then fed the other one from the load side of the first in a criss-cross, that would probably make things trip. But I don't see that changing the grounds would affect that.

Depending on the age and type of box you may be able to attach another device box to it. The old box needs to be an expandable type with screws holding the sides on. The dimensions for the screw placements have moved around over the years, so it could be hard to find a compatible box to attach. You will need to make a larger opening and do some wall repair to get the new gang in and attached. I would remove the old box and move the cables into a replacement 3-gang old work box. If done very carefully the hole in the wall for the new box will be the correct size and no wall repair will be required.

[i]"No reason why you can't curl the 4th wire for a bottom knockout."[/i/] I was thinking that running the 4th wire down from the ceiling, then behind the box, and then curled back up to enter a bottom knockout, well, isn't that going to mean that the 4th wire isn't going to be anchored to a stud? I guess it could/would be anchored most of the run down, but that drop behind the box, and then the curl up, that's neither gonna be anchored, nor is it gonna look as "neat" as the 3 wires that drop into to box from the top. Still, the question to be answered about the "drop behind/curl back up" method would still be: "is THAT to code?"

I won't be able to double check until tomorrow, but yes, it looked like it is on a 15A circuit, nothing about it made me think it was on a 20A circuit, it's on 14-2 wire (consistent with being on a 15A circuit), and it is in a bathroom, just powering a clock radio and an electric toothbrush when we don't temporarily plug in a razor or some hair trimmers. This circuit powers this bathroom's 15A GFCI, and then the line side continues to power some outlets in another room. (I realize that those outlets in the other room are not protected by the GFCI since they're coming off the line side instead of the load side.) So are you saying that all bathrooms MUST be on a 20A circuit? If that's true, it makes me wonder if the old home owner added this line himself and didn't know or care. I just added the two receptacles to the existing GFCI that was already there on 14-2 wire on a 15A circuit. But I'll double check tomorrow. Hopefully, if anything, it is actually on a 20A and 12-2, then I can just switch out my new wires. I just can't see how I would not have noticed that.