Showering the poly-cotton allows the cotton fibers to expand, fill any holes around the stitching and tighten the weave of the fabric. There is no need to waterproof the fabric after doing this. A note on this. If the poly-cotton has been coated with a waterproofing layer on the outside of the fabric then this technique will have no effect on the fabric. This wetting down technique only works on poly-cotton that has the waterproofing absorbed into the fabric itself. My article goes into depth on the advantages of fabrics that have the waterproofing, UV resistance, mold resistance, and color absorbed by the fabric rather than being sprayed on.
Regarding folding the tent away wet, there shouldn't be a problem with the bedding getting wet if the tent material had a high degree of waterproofing in the first place. The material of the tent folds upon itself with all the wet outer layers folded to meet other wet outer layers. The dampness is not transmitted to the inside of the fabric.
Hope that helps.