Haven't turned in my primary yet. Is early July too "late"? Will I be at any disadvantage from filing in early July compared to those who filed in June?
It won't be too late but you will be at a serious disadvantage because it will probably take about 4-6 weeks to get verified, so your app, at the earliest, will be complete in early august if you submit within the next week.
You should be fine sending in your application in the next few weeks (early July). There's a balance between completing/editing your primary and your submission date; however, it's certainly better to have a solid primary submitted a little later than a less solid primary submitted earlier.
You'll have the rest of July and the beginning of August to work on secondary applications. I HIGHLY recommend writing all the essays for your secondaries during this time so you can submit all of your secondaries within a few days of being verified. Depending on your schedule, maybe dedicate a few hours each day to writing 🙂