Welcome to GOY. I am sure that there will be lots of suggestions but in the meantime have a look at this site for some ideas. The weeping Katsura looks good - but expensive, though you don't have to buy from that site.

http://www.tree-shop.co.uk/products_group_list.asp?productgroupID=1&offset=15

I have a lovely weeping Larch (Larix) at my pond side. Seems to enjoy the wet soil, but check it first...sorry its not evergreen. Not many evergreen trees like wet boggy conditions. In fact I cant think of any off the top of my head. Hope somebody else will know better! :) i have bought trees from Bluebell Nursery & Arboretum online. They are excellent and would certainly be able to advise. Their website is great too! :)

I can't ever remember seeing a weeping evergreen on marshy round. Had you thought - if you have an evergreen you'll miss the pleasure of the new leaves (and blossoms depending on what you plant) in the spring?

True Stera...but a screen is required. I think a Bamboo might be best, but tbh I'm not sure about the marshy ground for that either. I know Alder is ok with marshy soil, but again...not evergreen.