If it's nearly trash already . . .
option 1: Several of the models have folded shapes with long edges. If you're careful, you can use the folds to pull the sheet back into shape. align the bow along the long straight folds, with the convex side pointing outward. To make the turn, it'll have to be straight along the fold. watch for warping down the length and alignment of edges beyond he folded areas -- you might have to scrap it anyways.
Also, If you're using Adam's foamboard (Which from your locale you're probably not, but others here do) *any* moisture on untreated paper will cause the sheet to separate from the foam. Here in middle Georgia the humidity alone is enough sometimes.
Which leads to option 2: strip the board. The naked foam is very compliant and will likely flatten on it's own with the paper removed. The stripped foam is about 4-4.5mm, and has about 1/2 the weight. It can be laminated for strength or shape with *WATER BASED* polycrylic (really like minwax's polycrylic , paint on, press & hold flat and give it 6-8hrs to cure), and can be recovered by packing tape, or coated w/ polycrylic and painted.
Now several of the methods in the swappables rely on the paper, so a stripped & tapped board might work for a particular swappable -- might not -- but this is only a *SMALL* subset of plans that can make a sheet of foamboard fly!