$34 for a 25 Lb. bag is a good price, if that seed is fresh (not more than a year old) you got a good deal. Your weather isn't much different than mine,
so planting in the spring doesn't give the plant's root's time to develop before the heat of summer, so expect some failure. Sept. 1 is the best time to plant. Aerating in the spring can be iffy because the soil and plant's aren't warm enough to recover quickly, just be aware of that. If you already have 50% grass growing, then you only need to seed at 3 Lb's per 1000 sq. ft. The number of plant's per sq. foot is important. Sunlight and air movement make the biggest difference in good growing condition's, do everything you can to improve that.
For your own information, you are in growing zone 7a, here's something to look at. Click on Tenn. Good luck.