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?

it's definitely not nutsedge, which has a different and distinctive look. first question to ask yourself:is it the exact same grass as the rest of your lawn? if it is, then extra water or fertilizer in that area is producing extra fast growth. If it's definitely a different plant, then you have a weed grass in there. I've had that. Big mess. Hard to get rid of. Good luck

My house was on the back of a 1-acre lot on the edge of town. The only tall grass was back by the house, so you had to stop and look kinda hard to see that the grass was that tall. If you were going to be offended, you had to stop and make a big deal out of it. When it's all the same height, it still looks manicured, especially at a distance. Well it turns out there was an ordinance against eyesore lots. The ordinance had words to the effect that you could not have piles of junk with weeds, grass, or plants growing 18 inches high. We all know what this ordinance is there for - so people will not toss their junk on the street and let grass and weeds grow up through it. I did not have piles of junk. I could walk down my street and find at least 5 other houses with piles of trash and weeds growing up through them for 6 months or more. The problem with my house is that it used to belong to a prominent lady in local politics and the church. She was one of those "GET OFF MY LAWN" people who took exceptional pride in the appearance of the yard. She passed away and it went to hell. A guy bought the house to rent the rooms and I bought from him. He let half the lawn die completely. All this happened so fast that the neighbors and friends of the original lady seriously resented my "letting the lawn go." So I was selected for one-time enforcement of the ordinance. I showed up at a city hall meeting and convinced them they needed to print a lot more signs and enforce the rules everywhere in town. They did that and quickly overwhelmed the enforcement employees. I also took issue with the word, plants, in the ordinance. Strict wording of the ordinance would prohibit any plants taller than 18 inches. Everyone has plants taller than that. I started mowing the lawn down to 17 inches to test my skills at passive aggression. A year later they rewrote the ordinance taking the plant/grass/weed height down to 12 inches. So I mowed at 11 inches. Mowing that high takes a string trimmer and actually goes pretty fast - much faster than mowing with a mower. So it was nice for awhile and then not so much fun, but I did learn a lot about lawn care.

Im guessing your lawn blades arent to thick so they are having a hard time springing up but I could be wrong. Either way, you can try using a leaf blower to fluff up the lawn before mowing it.