Moist soil will limit your options for most types of evergreen shrubs or small trees.
Is the water runoff from your property? Or your so-called neighbor's?
First thing I would do, is call the Zoning office in the county/city, to determine if the shed & compost bin is in compliance with local building codes/ordinances.
Depending upon the size of the building lot, there is usually a buffer required between any structure and the property line. In my county it is 10 ft. To build any closer to the property line, a side-yard variance must be obtained, that requires posting signs announcing a hearing date before the zoning commission. If approved by them. it then has to be heard and voted upon in a public meeting of the County Commissioners. If approved by them, then you must apply for and receive approval of a building permit. The permit must be posted in a location visible from the street/road, before any construction begins. At any time during these proceedings, neighbors have the right/opportunity to challenge/object to the variance.
Did your neighbor do all of those things? I assume he didn't bother or else you wouldn't be trying to shield the ugliness.
Call your Code enforcement office and have it checked. If in violation, insist that they be removed. Settle for no less!
I wouldn't be concerned about their feelings, as they obviously have no concern for yours!

Hope I'm not to late to add my thoughts here. Evergreens, with the exception of yew and hemlock don't fare well in shade. We have all-year-round semi-evergreen screening in a shady area with leatherleaf viburnums which form an attractive and dense screens with bonus of blossoms in early summer. I'm not sure about the moisture tolerance of leatherleafs but I do know many viburnums are.
Good luck. I have the same problem...needing to screen off neighbors unsightly "landscaping and building projects" just to the side of our driveway. Argh!