You probably have so much bare dirt because of the tree shade. The CG is likely growing in what areas do get sun. tiemco is the seed guy and he can advise you, but you are going to have limited cool grass options. You may have some warm grass options and besides tiemco there are a number os Southern gentlemen here that may be able to advise on seed. DIY tests aren't much use, you should consider getting a professional test. Logan Labs in Ohio does a very comprehensive test for $20 and will make recomendations for $30 more. If you post here, maybe someone will analyze it and give recommendations--not guaranteed as few can. Don't till. You'll want starter fertilizer. My opinion is that it is best applied at seeding. Pictures would help for the seed recommendations.

Thanks for the reply. I was thinking creeping red seed as it is supposed to be good for shade. If not tilled what would you recommend?
Iwas also thinking starter fertilizer but it is low on nitrogen and the test shows very low nitrogen.

I would suggest using a verti-cutter/slicer, a machine with fixed blades that make a verticle cut. Some rental places will call them power rakes, just make sure it has fixed blades, not tines or flail blades. It is very versitle and can be used to remove thatch and dead matter from the surface, it can be adjusted to depths of 1" for loosening the top layer of soil and can be used to slit seed. Stick with starter during the planting/early growing of your seed. Too much nitrogen can harm the seedlings. Wit until after at least the third mowing to start adding recommended rates of synthetic fertilizer. You can add organic almost any time (you could apply Milorganite prior to loosening the soil with the verticutter-- if you decide you need to loosen the soil).

The problem with tilling is the soil never settles evenly. It is especially uneven when you have rocks and/or tree roots to deal with. The tiller bucks around and digs into the soft spots leaving a very uneven subsurface. When you level off the surface, that fluffy soil eventually settles back to match the uneven contour of the subsurface. Your surface will become bumpier and bumpier for 3 full years. So the recommendation is to leave your soil structure as it is and deal only with the very surface. The vertical rake (cutter/slicer or any of many names) is the machine to do that.

Don't put chemical fertilizer on bare dirt. That is a waste of fertilizer. Unless you have roots there to absorb the fertilizer, nothing happens. Organic fertilizer is different. When you put that on bare dirt, it improves the soil. Chemical fertilizers work on plants. Organic fertilizers work on the soil...which then improves the plants.

Does anyone in your area grow St Augustine in the lawn? Tennessee is a transition state where people get by with northern or southern grasses depending on very local conditions. If you are on a mountain facing south, then grow southern grass. Facing north? Go with norther grasses. Some places you have more options.

You can soften your hard packed dirt by spraying it with any clear shampoo at a rate of at least 3 ounces per 1,000 square feet. The shampoo will allow moisture to sink deeper in the soil and hold it there while the beneficial fungi in the soil regrow. Once the population of beneficial fungi is reestablished, you will not have hard soil anymore. Two applications of shampoo at 2 weeks apart is usually all you need for at least a year. Cost is a couple bucks for a yard your size. My last purchase was Alberto VO5 at $1.00 for 15 ounces.