I've been randomly following this thread, and would the OP be better off installing a booster pump for the system to help get better coverage?
I understand there might be a design problem, but just thinking out-loud on the easiest fix for this situation.
That and one thing that stands out is the fact that the soil is a sandy soil type. From my experience as a farmboy and irrigating sandy fields is a tough proposition at best. Seems like as soon as our irrigation pivots made a full turn, you just went right back out and turned it on again.
So even with the best designed system, the lateral movement of water is next to none, and getting even growth across the lawn in extreme heat and evaporation days to me seems like a waste of time.