In that first picture it almost looks like those might be eggs of some kind? I had a few plants that got powdery mildew and they did not look like that. Sorry... not much help - this is my first year too, but like I said, that's not what my powdery mildew looked like.
How much sun do those plants get? I know that lots of things say full sun... and I've never lived in Las Vegas, but I know here in Florida, even the plants that say full sun don't like the full sun here in the summer. That is to say, really hot, desert, afternoon sun is pretty harsh on little plants. Just something to think about.
I also think that when I started my garden, I was over-watering... partly because of how hot it was. But I learned that the Mel's Mix really does hold moisture well. I check daily, by sticking my finger an inch into the dirt to see how moist it is and water if they need it. So maybe there's a little over-watering? But maybe not. Like I said, I'm definitely not an expert.
Good luck, though. I'm sure everyone else around here will have some useful ideas.

It is NOT powdery mildew. Ask me how I know...I've suffered with it on rose plants for years b/c I haven't yet found the magic organic approach that will work in the mostly foggy climate of the San Francisco Bay Area.

I would take a well-focused photograph of this plant to a local, independent nursery/garden supply store and ask them what it is and how you can organically deal with it. After all, you plan to eat these veggies!

I agree with sarahs eden that it appears to be eggs, but I'm completely unfamiliar with gardening in Las Vegas and don't know what kind of insects/critters attack plants in your neighborhood.

And, yes, some shade might be a relief to your plants. Mel has some ideas about that in the book, since he deals with Utah desert and started out with SFG in Florida (I think).

Cynthia

They're some type of white furry mites.
Just spray with soapy water and they'll be gone.
I had some this spring and a good soak with soapy water did the trick.

Will

Darn that stuff!
I've been lurking on GW for over a year, and I've just stumbled on this post which has compelled me to register.
Vegas4x4, you are not alone. I think the white fuzzy dots are endemic here in Las Vegas, as they randomly pop up on the stems of all my plants. They don't seem to do permanent damage, and are rubbed off easily with a fingertip. I don't think they're eggs, as they're not gooey or anything, and don't really have much substance. Here's hoping we find out what they are eventually!