As you have discovered, it is very difficult to have a healthy lawn in a shady, poorly drained area, and a suitable ground cover is a good idea. The presence of moss indicates that the soil in this part of your yard may also be compacted or is of low fertility. Moss does not necessarily mean that you have acid soil. Have you ever had your soil tested? You might want to do this so that you can correct any pH or fertility problems. A soil test is quite simple and inexpensive and will provide information that can save you time and money in the future. Few of the plants I can recommend will grow well if the soil is infertile. Contact your local Extension office about how to perform a soil test and where to submit your sample.
There are some shade-tolerant plants that will grow in moist conditions, and I would suggest trying out several types to see which do best for you before investing in a large number of one kind of plant. Part shade: If you have clay soil, Ajuga reptans makes a very good ground cover. Ferns of many types but especially the Cinnamon, Interrupted, and Ostrich fern will also fill in shady, moist areas. Gout weed, or Aegopodium, is a fast-growing ground cover for shade and competes well with tree roots. Some hostas will also work; Thomas Hogg and Fortunei are recommended for damp locations. Primulas, ligularia, and astilbes would be nice accent plants here, too. Full shade: Pulmonaria and Tiarella would work. Pulmonaria seeds freely, and if you plant a few, you may soon have many. I put in two plants about eight years ago and I now have a large area of pulmonaria in my shady yard. They have lovely pink flowers that turn blue in the spring and large attractive leaves in the summer. (I cut mine back after blooming to prevent more seeding and to keep the foliage looking its best.)