Another try and a new approach.
I abandoned the idea of spraying on an undercoat and went straight for my brushes. All of the base colors were applied using “overbrushing” (which is just a fancy way of saying “drybrushing, but with more paint left on the brush). Also, the paint mix was a little thicker than the “milk-like consistancy” I normally use for miniatures or terrain. The overall idea was to go light on the amount of paint drying on the piece at any given time. This prevented the warping and fraying issues encountered with the first test piece.
However, this left me with another problem. The gaps between the cardstock pieces were still clearly visible. The idea behind spraying on a dark undercoat is to cover up these gaps in coverage before applying the base colors. With no undercoat, I had to find another way.
The solution was a dark wash. Normally, a wash serves to bring out details, but this time it filled in the spaces between the cardstock pieces. The result was the appearance of shadows where the cardstock pieces meet.
The wash did darken the piece considerably. I drybrushed the base colors over the wash as a highlight to bring back a brighter look to the piece.
The end result is a nicely weathered looking shack suitable for a “lived-in” sci-fi setting. I've got some plans to use a settlement on a distant world as the backdrop for some upcoming games, so this piece will see time on the board in the future. I might even make a couple more down the road.