"Not the best" is really not a good enough description for the soil you have available. Probably what you have is layers and layers of soil additions put in by various gardeners over time. We found this link to Brooklyn College which can do soil testing for you. It would be a very good thing for you to know what ingredients make up your soil and what is missing from it for healthy plants. We are going to introduce you to our Native Plant Database so you can select plants on your own. The Lady Bird Johnson Wildflower Center, home of Mr. Smarty Plants, is committed to the growth, propagation and protection of plants native not only to North America but also to the area in which those plants are being grown, in this case, Kings County New York.
Follow the link above to the Native Plant Database and, using the Combination Search, select on New York for the State, "herb" (herbaceous blooming plant), under General Appearance, "part shade" (2 -6 hours of sun a day) under Light Requirements. Then, since you are looking for a groundcover, go down the Combination Search list to find Height, and specify 0-1 ft. We tried this search, got 63 possibilities and took the first 3 that we liked. We would next like to mention a sub-shrub that is native to New York, Arctostaphylos uva-ursi (Kinnikinnick). It grows 1 to 3 ft, so it couldn't be walked on, but it is evergreen, a very important characteristic in New York. Now, you said that grasses died in the sunny spot in your yard, so we'll put in the same kind of search checking on "Grass/Grasslike" in General Appearance. Here are our results for suggestions for ground covers:
Ground Covers for Brooklyn, NY:
Follow each plant link to our webpage on that plant to learn its Growing Conditions, soil and moisture needs, etc. Since you may encounter some difficulty in finding native plants, go to our National Suppliers Directory, put your town and state or just your zipcode into the "Enter Search Location" box. This will give you a list of native plant nurseries, seed companies and consultants in your general area. All have contact information so you can find out in advance of a shopping trip what they have for your purposes.