Good catch BPGreen....in zone 7, you may be in a trasitional zone....that is, in an area that is in mid-area of the U.S. where turfs can be warm season or cool season type grasses. The area comprises mid-America. Before you decide which type you should use, speak to your neighbors, ask what they use, what they've had good results with and then speak to your local full service garden nurseryman who is up on the turf grasses that do well in your area. After all, he's in the business of selling to the public what sells. As to your need for a shade lover, why not speak to your neighbors about possible relief from the trees giving so much shade. A simple pruning of some branches can offer some additional sun. If you have trees that contribute to the shade, then pruning lower branches could invite more sun to the area around their base. Trees naturally steal moisture from grass around them and where you can increase such moisture it can only help.

A picture would be worth a thousand words here. Bringing in soil might be good if you suspect the original soil has been washed away for some reason. Some trees will react to hard soil by sending their roots out at the surface. Then as the tree ages, the roots become larger and expose themselves above the surface. If that is what has happened, then adding soil on top of the natural grade will cause you drainage problems and could cost you extensively if water pours into your basement (worst case). Adding soil should be done with considerable forethought as to the consequences. Another thought on adding soil is to keep it lower than the root bark on the trees. Tree roots flare out from the base of the trunk. If you bury the root flare completely, then you are burying tree bark. Tree bark will often rot when it is below the soil level. It is best if you leave some root flare above the soil level to be sure you are not burying tree bark. Tilling is a bad idea for many reasons. One of the obvious ones is that you have roots everywhere. You can't till tree roots. A much less obvious reason is that tilling destroys soil structure by fluffing it up. Once it is fluffy it has to settle again. That process takes 3 years. When it finally settles it will settle unevenly leaving your surface bumpy. Grass seed should never be mixed in with surface soil. It always goes on top of the surface. Many people recommend "scratching" it into the surface but that is not necessary. Nor is covering it with straw. All you need to do is loosen the soil at the surface (1/16 inch deep) before you apply the seed, then apply the seed, and finally roll the seed down with a water filled roller (rented). Seed should remain at the surface. If you insist on straw, as goren said, be careful not to use hay. Hay is forage for animals and has seed in it. Straw is the left over after the seed is threshed out. New seed must be watered 3x per day at breakfast, lunch, and dinner for about 10 minutes. The idea is to provide enough consistent moisture for the seed to swell and open (germinate). Unfortunately these are the exact conditions that crabgrass seed needs. For this reason grass seeding is most successfully done in the fall after crabgrass has stopped germinating. Also grass sprouted now will have weak roots that cannot stand up to the summer heat and dry weather. That might not be that much of an issue in your area. Don't assume you have clay. Fill a mayonnaise jar half full of dirt. Then fill the jar completely with water and a drop of soap. Swish it around and let it settle. You will see rocks and sand settling immediately to the bottom of the jar. In an hour all the silt will have settled. At that time if you have clay, the water will be cloudy. Let it sit another day and see if the water clears up. If it clears then you do not have clay. Clay will cloud the water for days and days. There are several conditions of minerals that cause soil to seem to be clay when in fact there is no clay at all. Don't be too surprised by this. Adding gypsum without doing this test and a soil chemistry test can be a mistake. I disagree with goren on mowing height. The only grasses that need to be mowed lower than 3 inches are bermuda, centipede, bentgrass, and some zoysias. Kentucky bluegrass (which you can't grow in shade) should be no lower than 3 inches. Fescues and all the other grass types should be mowed back to 4 inches. Usually this is a mower's highest setting. Grass grown in the shade needs to be longer simply so it can photosynthesize. Can you post pictures showing the tree roots and one from a distance showing the trees with the canopy?

Check with Moss Acres on the net and see if they will let you use one of their images in exchange for a little promo on their business.

Luzula, Chasmanthium, Deschampsia...I'll have to think of some others...