The timing for harvesting the plants is like with the millet, if I'm using it for seed: The finches and chickadees will let me know.
They will start pulling the outside seeds from the sunflowers. Those are ripening first. It is a good idea to have them someplace that the birds can't get at them after they are cut but they probably only need to stay there for 2 or 3 days in your dry climate.
(If I see that the finches are eating the millet - I've got less than a week before they will take every seed.)
Hardware cloth screen (1/2") might be a good way to get the sunflower seeds off the heads but I haven't done that. Gloves work fine.
I had to use my hands!, they are easier to pick when slightly dried, dont over dry it, the seeds wont be good !! Just rub them off the flower into a bowl !
Here's something that I use to keep the birds off the flowerhead until I can harvest. I buy aluminum screen at the hardware store and cut it up like I would cut up fabric for sewing. I make little "hats" to put over every head that I want to harvest the seeds from. You can make your own patterns and you can use fishing line for thread to stitch the pieces together. I have a few that are 6 years old so they are pretty durable. The alum. is light weight and I always try to make my little hats as small as possible to save weight. They don't have to be pretty (at least for me) they only have to be functional.
Using my "hats" allows the seeds to mature naturally on the plant until they begin to dry on their own. I harvest for eating and for seed collection.
One year I tried the black or grey fiberglass screen and the bigger birds were able to tear it up. I'm sure they were starlings.