Core aeration is a temporary solution that might help for a short time. The long term solution to soil compaction is adding adequate amounts of organic matter to the soil.
Most turf grasses do need sun for growth but there are some that can grow in shady provided the soil they are trying to grow in is a good healthy soil. A good reliable soil test is needed to determine if the soils pH and major nutrients are in balance so the grasses have a chance to grow.
Japanese Stilt Grass appears to grow nest in soils that are acidic (more acidic then desirable grasses like), moist, and shady. Changing 2 or more of those conditions may be all that is needed to eliminate this invasive plant.
Moving sod means stripping it with enough of the root system attached so the sod can survive in a new place. That can mean a minimum of 4 inch to maybe 6 inch depth of cut.
Check with your local office of the University of Maryland Cooperative Extension Service about soil testing for soil pH and major nutrients and maybe these simple soil tests might be of some use as well.
1) Soil test for organic matter. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. For example, a good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top.

2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drainsâ too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer your soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy.

Here is a link that might be useful: UMD CES