I run Advanced Quality Lawn in Northeast Ohio, and drought management is something we deal with in real time across lawns, trees, and shrubs in a lot of different soil conditions and zip codes. The biggest wins have come from changing the *watering habit*, not just adding more water.
The single conservation practice that's had the biggest impact is deep, infrequent watering instead of light, frequent watering. In our area, I'd rather see a lawn get the water it needs for the week and push moisture down into the root zone than keep the surface constantly damp and train the grass to stay shallow-rooted.
A simple example: when we walk a property during a hot dry spell, we don't just ask "is it getting watered?" We check whether water is actually reaching several inches down, and if it isn't, we adjust the duration rather than adding extra short cycles. That usually improves drought tolerance while also avoiding the overwatering issues that can invite disease and pests.
The other part that matters is mowing correctly during drought stress. Keeping the lawn a bit taller helps shade the soil and slow evaporation, so the water you do apply stays useful longer.