I would use compost or top soil instead of sand.

Adding a half inch or less of compost to the top of the grass and raking it in will improve the soil. Do it every year and in a few years, you'll have very good soil.

The areas that dry out faster in your lawn, they might have a rock, or other non-soil material just under the top soil. Push a long screwdriver into those areas and see if you hit any resistance near the surface. I have a friend who has a concrete path that the builder buried in her lawn, and in times of no rain the area forms a perfect dried out rectangle.

Yes, you can use sand. Use sharp or masonary sand rather than the bagged play sand as play sand is often washed in chemical. I add peat, peat moss or compost into the mix along with the seed and I don't mix anymore than what I can use in a couple of days.