Ive moved 100's of games in the rain, literally hundreds. Go to rural king, famr and fllet, home depot, or somewhere like that and get a tarp. Lay the tarp down on the gound, then dolly the game to the center of the trap. Fold the tarp up on the sides as high as you can go, then shrink wrap the hell out of the game. being sure to go over the top real good.
You mentioned worrying about the wrap comming off with the drive. To prevent that, take a roll of packing tape, like the 2 inch wide kind. start at the bottom and just walk around the machine so that the tape goes about every 18 inches, this will keep the shinkwrap from being able to be frayed and coming off. If you do the game like this, not a single drop will get on the game.
Like a few others have said, once you get to your destination, remove all the wrapping.
The key part that i mentioned that others have not is by placing the game on the tarp you are covering the bottom. The bottom is actually the worst part that most dont cover. When you are driving, the water from the street actually flies up and back into your vehicle because of the air draft.
Lastly, if your game gets wet, dont sweat it. The game getting wet is not what hurts the game. The game staying wet is what does the damage. always wipe your game down when you get done with the trip to get the moisture from humidity off.
troy