You don't need to have a fan blow hard over it, just hard enough to recycle the air which is not necessarily hard at all. The mushrooms will go bad if the drying is not fast enough plus there are lots of germs in the air and its just humid, dirty, moldy right there. That's a bad recipe. If you dry fast enough you are not giving those a chance.
I'd avoid paper towels, they stick to the mushrooms when they dry and they are simply not necessary. Instead use gauze like material to allow air to reach the mushrooms from all sides. Alternatively you can turn them over every so often but it's not recommended.
What I do recommend is starting with some air drying for the most moisture to get out, then finishing to cracker dry with the desiccant. The desiccant should not touch the mushrooms, they only have to be in the same enclosed airspace. This is then a desiccator. The dessicant dries out the air and the dry air dries out the mushrooms so to speak, it's an indirect system. The reason you need to recycle air with a fan with air drying is to use new air that is relatively dry again. You cannot dry something with air that has already taken up water from that something.
If you cover fresh mushrooms in the fridge maybe the problem is condensation that can drip onto them and get them to spoil. On the other hand when you were able to buy fresh psychedelic mushrooms from smart shops in my country before they were banned they were in an air-tight plastic package so whatever.
You should not try to preserve mushrooms in a fridge anyway unless you plan on consuming them within a few days tops. Otherwise dry them!
What does the mushroom growing kit have to do with preserving or drying them? The growing kit is for growing only, what you do with them after harvesting is the next chapter.
Anyway you should start learning how to use a search engine. Almost all questions you ask are very basic and answered very well in tons of places. Google is your friend.
Lastly - I don't mean to be a partypooper - just for info, HydroShrooms is not as innovative as it seems... in my opinion. Normal shotgun growing chambers often involve Perlite, which I used as well. It is a sort of porous mineral that can contain a lot of water, just like the hydroponics beads. So in the end it's just all a nice way of gradually regulating a constant high air humidity.
I do think it's pretty sweet that it includes air filtration. If you can afford it, I guess this is a nice solution and it allows you to grow without really understanding what you're doing. No offense again, but you just sound more result-oriented and in a hurry than wanting to become a growing craftsman.
I wish you the best of luck with the grow: hope you get a satisfying result and a good trip.