I have to start by saying that wet areas should be allowed to stay wet.
That said, there are some plants, many of which are non-native and invasive, that will do the job -- like weeping willow. Red osier dogwood likes wet areas and is eco-friendly. Likewise for Jewel weed.

Patrick, I do believe that dogs like it wet as do elders and even native hollies. Where are you in NH? I am on the MA/NH border, so am curious.

I wonder if the natural progression of build up of organic matter from the plantings will serve to "dry out" the area in a way? Plants certainly play a huge role in erosion control. I know nothing about this, but just wonder. I notice that there is a natural evolution to such areas. I too believe that a wetland should stay "wet," but the addition of plants and then, eventually, their organic waste might serve as a sponge to keep the area wet but not soaked. I believe that swamps are good for that purpose and serve a very important role in the water cycle and health of the surrounding environment. Just a thought.

We have native hollies volunteering in our wetland. We also planted duck potatos and some other native plants. If the area gets some sun, Joe Pye Weed will do well in a soggy area. Native witch hazel seems to be able to handle life near a stream and in wet conditions. The red osier dogwoods really love the wet areas and has made thickets.

Don't know about what is native to your particular area, but I would add birches, cottonwoods, and sycamores to that list, as well as pawpaw.
Then there's bladdernut, which is a water-loving, lower-growing shrub to add to the understory.
Again, might not be native for you.

Is this an area that has soggy tendencies because of human intervention or does it come that way naturally?

I forgot to mention that elderberries are also doing well and expanding in our wet areas.