Desert Camping?

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To begin with, I am planning on going camping for a week in desert areas all rural, no restrooms/showers etc. I am supposed to get my period during that time.

I'm worried about the leaking and disposal of products. I use a cup, and it does leak easily (still have not figured out why). Since I will be the only woman on the trip, that I know of, I really have no idea what to do..I can't talk about it with any of them.

With the heat and the amounts of hiking I really don't trust the comfort of pads and don't want to end up smelling funny...I am considering cloth pads, but then I would have to wash them and really I don't want to deal with that...I would love to be as inconspicuous as possible.

I've been trying to find some ideas online but it mainly seems to be asked about in the woods camping and not desert conditions..So, any friendly advice?