"Do I need to let the entire bloom completely dry on the stem before I cut off a zinnia bloom to save for seed?"

No, you can save seeds as green seeds as long as they are filled out. You can pluck a few of the bottom seeds to check on whether they have developed viable embryos.

If a zinnia head contains some "fat" seeds, even though they are green, and the petals still have some of their original color, you can dry those inside.

You can dry a whole seedhead, or your can pull the petals out to see which ones are not "empty", and just dry the good ones and discard the rest of the seedhead.

There are actually advantages to harvesting green zinnia seeds. You avoid rain damage which can cause zinnia seeds to sprout in the seedhead (which ruins them), and you get them inside before the seed eating birds, like finches, eat them.

ZM