Spring isn't the ideal time to plant Fescue, which is a cool season grass, as it doesn't have time to establish a good root system before the heat and lack if rainfall occurs during the typical summer season around here.
Probably wouldn't do much good to overseed the lawn now, best to wait until Sept or early Oct to do that.
Best method for overseeding is one recommended by Walter Reeves. He suggests renting a machine that areates the soil by removing small plugs. broadcast seed, lime and fertilizer at the recommended rate, then go over the area with some sort of drag to pulverize the plugs and cover the seed. He suggests that a section of chain link fence, (4'x6' maybe easy to handle), is ideal if available.
Then, water it well and keep the soil moist until the seed germinate. Next spring, you should have a great lawn, ready for mowing, fertilizing, liming, mowing..mowing!

What do I do with the dead grass though? Do I pull it up or just leave it for the fall?

It appears you have a good deal of live grass. Good job!

I'd recommend watering the lawn one inch a week AT LEAST....perhaps every five days if we get more brutally hot weather. And set the mower high - 3 inches, I think. That will help moisture retention in the soil.

This fall, get recommendations from a good garden center as to when and how to overseed. Also, check to see what Walter recommends at georgiagardener.com.

HTH, Rosie

You can scalp the grass and overseed in the fall or around mid september. I gave more suggestions in your post in the lawn care forum.