Yes, COBRA will cover you, but you have to pay for it (retroactively if you need it). You've already paid for your company medical plan through your payroll deduction, why not get one month for free? Basically, your company supplements the cost of health insurance, usually somewhere around 80% of the group plan rate - that is about the average. So the $50 a month you get deducted from your paycheck tax free goes up to ~$250 a month post-tax for COBRA. Just work one day in August and your covered the whole month for free.
If you leave Aug. 3rd then you still have 2.5 weeks before KWEST, plus free health insurance till September. If you leave July 31st, then you you have 3 weeks till KWEST, but might have to retroact COBRA (and pay $250) if anything were to happen to you in the month of August.