Thank you! I had gotten a response from some one on the bulbs forum who suggested that the papery layers may just be a bit bigger than the bulb, with air between them. It seems like I had a bigger risk of the bulb being dried out, rather than rotten, as the sqishy-ness was due to air pockets, rather than a moist bulb.
Per her instruction, I carefully inspected the bulb and it looked to be healthy, so I have planted it and will see how it goes! Unfortunately, the package just said "Elephant Ear - Colocasia". Ha, it was a random purchase at a Fred Meyers, but my mother-in-law wanted to split the package, so I took a couple home. I am crazy about aroids, so I will hopefully be able to ID it once it sprouts and shows me its leaves. If not, I'll post it on here for help IDing, everyone here has been so quick and helpful so far!! I have learned so much already!