I’d recommended that you do this project in advance of using it as wrapping paper since you do want to let it dry. After I sprayed my papers, I hung them on my clothes line to dry out. The packing paper will dry fast. This will also help remove that spray paint odor. I also ironed my packing paper when it was dry.
Of course you can make papers like this as well to use in your mixed media art or journaling/collage.
I also made my own ribbon with torn scraps of muslin fabric that I scrunched up and ink sprayed. When they dried (also clip to clothes line), I ironed them and then did some marking making with a black sharpie in spots that I felt needed some interest. You can also mark make on your papers in the same fashion.