Answer: I don't think the heavy shade is as much of a problem as the soil you're working with. Some great groundcovers for heavy shade include: Sweet violet (Viola odorata), Black mondo grass (Ophiopogon planiscapus 'Arabicus', Japanese painted fern (Athyrium nipponicum var. pictum), Lily-of-the-valley (Convallaria majalis), Pachysandra (Pachysandra terminalis), Wild ginger (Asarum canadense), Holly fern (Cyrtomium falcatum), Cast iron plant (Aspidistra elatior), Confederate jasmine (Trachelosperum jasminoides), Vinca minor, and Asian jasmine (Trachelosperum asiaticum).
With a sloping area, the water will naturally run off before it soaks in, unless it has somewhere specific to go. If you take the time to amend the soil with organic matter each time you install a plant, the water will tend to soak in rather than run off, especially if you apply water slowly, either through a drip system or a soaker hose. If it is impractical to amend the entire area with compost, peat moss, etc. prior to planting, at least dig some of the organic matter into the planting areas and leave a small indentation (watering basin) around each plant to help channel the water to the root zones. Then if you apply water slowly for an extended period of time it should soak in rather than run off.
Best wishes with your garden!
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