How should I deal with a full ground bus bar in an electrical panel?
Seeing that this is your main panel the ground bar, neutral bar and the metal of panel should be bounded, usually with a bonding screw, in some way. So putting the new ground wires on the neutral bar is okay. Now with that said,  if you don't want to do that, why not add another ground bar if you have the space?  If you look at the sides, top or bottom of the panel you may see holes (with a ground symbol) for another ground bar to screw down to. If you see this turn off the power and add the bar.