I live in Northern AZ. You're likely to see almost daily afternoon thunderstorms in July. June is the scorcher here. The trick is water. You would need more in June and some of your path is pretty desolate country. Just for scenery, I would skip Farmington etc. and go through Cortez, Bluff -long stretch through Monument Valley - Kayenta, -long stretch I'm not familiar with- Page, Kanab, etc.
PS. You'll be going against the wind.