Here's one more pic of the leaves and spots.

update:

I took the pics to a local nursery, and the workers there said it looks like a fungus. Does that sound right?

And if so, any recommendations on how to treat it?

The workers there suggested I cut off all the affected leaves OR use a fungicide (which would have to be safe for vegetables, as these are planted right next to summer squash and cucumber).

Sunflowers often have ugly leaves, still usually make great flowers. I wouldn't treat annuals with anything at all, once they set seed they will die anyway. A sprinkler-type watering, that gets leaves all wet, can aid/spread these types of pathogens. Is that a factor?

Removing the affected leaves could help it spread less quickly. Cinnamon and milk can be natural fungicides, against some pathogens, you may want to investigate those if you'd feel better by being more proactive.

Thanks for your response.

We have drip irrigation, so only the roots are getting water. And it hasn't rained here in months (high desert of Southern NM), so that's not the problem either. I was surprised when the nursery worker thought it was fungus, as it is very dry here. But I was watering in the morning and evening, and she thinks the evening water was creating a lot of humidity, as it 'steamed up' from the hot ground.

I'm going to remove most of the affected leaves, leaving only a few that are least affected (as I don't want to completely defoliate the plant). Then I'll try a natural fungicide - thanks for the suggestions of milk and cinnamon. I'll check into those.

The creating humidity thing sounds very likely, that it would happen and that it would exacerbate a pathogen. Interesting, advice I could probably heed also, make more of an effort to get any watering done earlier in the day.

With no experience growing anything, or even being in a desert, I'm at a loss, apples'n'oranges. But watering every day is something I would question, at least in a less dry place. Have you tried a slow, thorough soaking, then waiting for about a week or at least a few days?

Do you have any material you could use to mulch the base of these plants so the moisture isn't lost so quickly?

You may want to solicit advice about growing sunflowers in the desert on the annuals forum and/or AZ forum, if there isn't one more specific to your state or to desert gardening in general.