OK, I figured out how to get the doors open, so I'll post this for anyone who may come across this same situation. I figured out why the passenger side mechanism was sort of floating in the door panel, which lead me to figure out how to open my rear doors. There is a screw on the other side of the door handle where it pivots. This screw sort of holds the mechanism for the handle from coming unclipped or retained on the opposite side of the handle where the release is. I ended up pushing on the mechanism to clip in the rubber bumper and the entire thing fell over and down, but still easily reachable from the hole in the door because the screw had slid out of its slot (apparently it must have been a bit loose). I was able to work the screw back into position and loosened it some more to slide it back in place. While it was pushed forward a little I installed the rubber bumper for the door handle that was removed when I took out the handle in the first place. I then slide the screw all the way back into the slot and the opposite side latch or clipped into something because it was now locked in solid.
So using some reasoning, I figured I would try loosening and releasing the mechanism on one of the rear doors. After the mechanism was released and fell forward and down a little, I was able to use the inside handle to open the door. This worked for both rear doors. Now all the doors are open and working properly.