There are many plants and shrubs which will survive and growing happily in damp shade. These plants which will tolerate partial shade conditions, and ground conditions which are damp, as opposed to plants suitable for dry shade.
There are three examples illustrated above. Left is Digitalis purpurea, Foxglove, of which there are several varieties, including a pure white variety 'Alba'. Digitalis will grow in sun or shade and will grow where it is damp, and also in dry areas such as walls, Foxgloves are a very forgiving plant. They are biennials which flower in the summer and self seed around the garden to produce new flowers year after year. Digitalis are easy to grow, but one cautionary note is that all parts of Digitalis are poisonous.
Illustrated above right is Hosta, well known for being tolerant of damp and shady conditions, some varieties more so than others. Ideal Hostas for damp shade are H.fortunei 'Albomarginata' large Hosta 75cms with green and white ivory variegated leaves; H fortunei albopicta large 75 cms variegated hosta ; H.'Frances Williams'(sieboldiana) smaller 60cms blue green leaves; H. 'Halcyon' smaller Hosta at 30cms blue leaves. Hostas are easy to grow (apart from the problem with slugs and snails;) and there are many many varieties to choose from of all sizes and variegations, with more tips and images on the growing Hostas pages.
Illustrated centre is Fritillaria meleagris, the delicate Snake's Head Fritillary, a spring flowering bulb happy to grow in shade and quite wet soils, see below.
Many of the plants listed below are easy to grow and are labelled with a green wheelbarrow, more difficult plants are labelled with an amber wheelbarrow which means medium difficulty to grow.