How to prevent damp items from getting gross in luggage

When driving, I don't pack wet/damp things except to get them to the car, instead spreading them out in the back.

But in general, it's far easier to have spares for travel that you bring back again. Then at least to go there you can pack dry; they may still be damp coming back. Toothbrushes especially are consumable and I keep a spare in my wash-bag, but hair brushes aren't expensive, and, while I'm not sure what you're referring to by "shower shoes", flip-flops* are often used for that and are the cheapest footwear you can buy. They also wear out, so getting a spare pair isn't even wasteful. In fact, having two of some things on the go can make each one last longer, as giving them the chance to dry out stops them spoiling.

To get clothing drier when at a hotel, try wrapping it in a towel belonging to the hotel, and trampling on the towel. This will help with the shower shoes, even if the towel is slightly damp before you start.

* _{Also called jandals, thongs, etc.}

Attach the wet items to the outside of your bag to dry. Use easy to attach and remove clips such as carabiners.

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Larger items like shower shoes (flipflops/thong sandal) and maybe the hairbrush, can be attached individually. If your shower shoes have a wider, over-the-foot band that doesn't fit in a carabiner, use a strap to secure them. A band of hook and loop tape (the kind with the hooks and loops on opposite sides of the same strip) will work for this, or tie loop at either end of a short piece of rope, and attach the loops with a carabiner.

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Put smaller items in a mesh bag or pouch. Attach the mesh bag to the outside of your bag. (Make sure to pack any private items in a separate pouch, and put that one inside your bag before carrying it in public.) Mesh bags come in a wide range of styles and sizes, and you can find very inexpensive ones at dollar stores and the like. Look for a bag with a sturdy loop to attach your carabiner. Or you can even repurpose a mesh produce bag, like the kinds that you get oranges or potatoes in.

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An advantage of this method is your gear is all attached together, so you don't have to unpack it when you get to your car, and repack it at the end of the day. The downside is that your gear will be rather floppy and rattle-y, and you may get some strange looks if you're walking through a public space like a hotel lobby. Then again, people may admire your ingenuity in finding a clever travelling/packing solution.

What you can do is buy some silica packets and put them together with the object inside a plastic bag. This way the items won't make everything wet and the silica will absorb the moist.

Since it is your car, if you leave the shoes in the trunk (not sealed nor in a bag), after a couple of hours they will be dry.

Brushes, simply wrap them in a towel and carry them in the space above the trunk. They will be dry, or at least they will not stink.

Consider shoes that have no cloth - something like a Jandal, Flip Flop, or Thong. Even Crocs could be suitable. Being hard plastic they won't absorb any water, and should get 80% dry by merely flicking them. A wipe with your towel should see them completely dry.

Carry a couple of tooth brushes, and dispose of the old one before leaving rather than taking it with you.

Try using a hard plastic comb instead of a hairbrush. Sure its not as good, but its dry and you can save space. I had a stainless steel one and it was awful at combing, but cheap injection-moulded plastic ones are ideal.