I use the plastic edging around a garden that's on a similar slope in my yard to keep sheet water from flowing through and washing out the mulch. Given your neighbors' yards appear to be draining through the area and the amount of rain you get in NC, I'm betting you have a lot of water rushing through there at times. BUT if you completely blocked off the whole area with plastic edging, it would flow right over and do absolutely no good (not to mention ticking off the neighbors in the process). Therefore, you could divide the area into three or four beds, leaving at least a couple feet in between to allow water to rush through the grassed or rock channels. You can then mulch the beds without too much water running the gardens and washing out the mulch. I don't know how much water comes through that area, so I'll leave a necessary size of the beds and channels to you. This may not be the most aesthetically pleasing way to do this but it may help keep the mulch in place. If you do go through with this, be sure to watch the edging you buy. At least one big box seller with orange aprons, in addition to its cheap stuff, offers a "professional grade" in a larger bundle that's of much much better quality (taller, thicker, heavier, more rubbery) for not much, if anything, more. In addition to lasting longer, it's a lot easier to install IMO.

To be sure, the wet ground situation is a completely separate problem from that of screening the properties. If it's a correctable situation, it's best to correct it and be able to use typical landscape plants, rather than being forced to use only plants that survive wet conditions. However, if you can't correct it, that's what you'll need to do. There's not information in the pictures or verbiage that suggest what might be causing the problem, so you're probably going to have to consult with a drainage engineer/landscape architect on site if you're going to determine what it is. It could be an issue that occurs outside of one's own property. The picture is meant to be an inspiration of possibilities, rather than a suggestion of exactly what you should do. To begin with, a planting scheme must be worked out in a measured PLAN VIEW (like a map of your yard.) That's the only way that you can determine how to arrange plants, how close to put one thing to another, and calculate quantities. The concepts I'm suggesting in the picture include taller plants at the yard corners, using groups of like plants as opposed to individual plants, knitting the border plants together with groundcover (it would be impractical to mow grass around a lot of separate plants ... and grass is not going to grow well in the shade of trees.)

"They [low priced nursery/garden center landscape designers] are used primarily as a marketing tool to sell plants, often very common or less desirable plants, and what you receive design-wise may be very substandard." My sentiments exactly. One would likely get the typical "builder's plan" ... the kind of problem prone landscape that everyone can't wait to redo as soon as possible. The forum is good for giving ideas about general direction to take, or spot problem solving ... but it does not produce PLANS, which are essential for a quality landscape product. One has the option of drawing their own plans if they are DIY oriented and feel competent about it. (They can get advice from the forum on the mechanics of doing it. ) If not, as in every other project, they must hire a professional. But landscaping without plan at all will probably result in something less desirable than a 'builder's plan.' The approach one takes for a back yard depends largely on one's goals and objectives. Some people want total privacy, as in a private paradise ... they don't want to see any part of neighbor's houses. Other people want a more open solution and prefer not to block much view between neighbors' yards. Still others want to block sight lines between theirs and neighbors' outdoor living spaces while keeping a feel of openness at other places. One needs to decide on their goals in order for a plan to shape up that would meet them. In nearly every discussion of creating a privacy border of some type comes the advice to avoid a uniform hedge. I would take this with a grain of salt because often, especially when space is at a premium, it may be the uniform hedge that best fills the bill. Many people here are plant loving "gardeners" and can't conceive of a planting that is not crammed with as much variety as possible. A potential downside to that is more labor in upkeep, more space consumption, and a busier look. If that fills the bill, no problem. But there are advantages to uniformity in potential space conservation, a cleaner look and simpler maintenance. Landscaping is not 'one size fit all.' Develop your goals and set about to see how they can be realized.

Landscaping a backyard for privacy means placing tall, screening shrubs near the offending property lines ... not close up to the house and patio. (There, you'd want to use low plants or overhead trees, or some combination thereof, between you and the view.) For the screening shrubs, you'd need to account for the varying light conditions, which means using a type of plant that can accept sun or shade, or using a variety of plants, each which accepts the conditions in which they are placed. Since you have a length of property to screen, it makes sense that you would place all the shrubs in a bed and keep them separate from the lawn. If you wished to incorporate any small tree forms in the mix, you might want to place groundcover below them instead of only mulch. (It's an opportunity for adding another element of interest. It could include groundcovers that bloom.) Since you want to screen the CL fence, too, it will be an L-shaped bed. If screening both CL fences, it would be U-shaped, following the fence and lot lines. The bed depth, front to back, should be sufficient to accommodate the shrub dimensions, which is going to be quite large ... so probably 8' or 10' depth. Understand that whatever ideas you are given and wish to contemplate, and possibly incorporate, need to be put in the form of a plan before you do any actual work. So prepare a base plan on which you can work out the ideas. You could begin that by tracing over a plot plan of your yard. If you don't have that, measure and draw out on graph paper.