Critter Bill Themm devised an ingenious (or at least as ingenious as Critter gets) raincoat for his bag which I am passing along...
Start with a plastic garbage bag ( one of the thick "Contractor" types), place your disc golf bag inside of it and start building side walls (just tape around the perimeter in circular layers) and a bottom with Duck Tape. Leave enough space between your bag and the trash bag so there is an inch or so gap on the sides and the bottom. Roll the top of the trash bag down and create a top hem out of more Duck Tape. You want the top of the plastic bag to be much higher than the height of your golf bag so the plastic can be folded over to keep rain out.
The gap between the plastic bag and your bag allows you to carry extra stuff in that space (towels, rain jackets, etc.).
Now punch holes in the plastic bag at the points where your straps connect to the bag. Put the straps outside the plastic bag and attach to your golf bag through the holes. Reinforce the holes with...more Duck tape. By attaching the straps through the plastic bag, the straps keep the plastic bag in place.
I watched Critter play in the rain with this contraption and it appeared to work well.