Do you have a photo of the area in the growing season? Or can you describe it as far as size, type of shade, slope, etc? Are there areas you don't want these plants to spread to? Are you willing and able to try to improve the soil some by adding organic matter as a mulch or top dressing? What type of area do you live in - coastal, suburbia, rural, and are there neighborhood rules that limit what you can grow? What zone are you in, since NH ranges from zone 6 along the seacoast to zone 3 or 4 in the north and mountains? (see link below if you don't know.)

I don't know how many dogs or how large or how the dogs use the area (just one area for potty, do they dig, do they patrol along a fence, etc.) but you may want to consider just mulch if they are likely to be very hard on the plants.

Have your tried growing a grass mix for shade? In many ways, grass is the least maintenance. Some plants that are good spreading ground covers for shade are periwinkle AKA Vinca major or V. minor, Pachysandra, and Carex 'Ice Dance.' All are evergreen and low-growing. Any of these will need weeding to remove tree seedlings a couple of times a year (pliers work well.) You can plant daffodils among the ground cover to give you spring flowers. Hostas may work as another plant to add to provide variety to the groundcover.

Regardless of what you plant, you will need to keep the dogs out of the area for the first growing season, so I would choose one part of the area to fence off and plant each season. Mulch well between plants and water the plants in well. Water if we go for more than a week without rain for the first growing season. After that, it should be fine without water unless we go at least 3 weeks without rain as long as you have it well mulched.

Here is a link that might be useful: NH zone map

I have a small side yard where I want to establish a small lawn and it has conditions similar to what you describe. Mostly shady, sandy well-drained loam, and on top of that the previous owner had a shed there which sat on several inches of crushed stone most of which had worked itself into the soil. Plus this little yard has 5-6 Norway maple stumps from previous overgrowth. Fortunately i do not have the dog problem. Sigh.

I dug out and screened 2 fairly large piles of rocks. I filled in the area with extra loam, then spread compost and lime in the fall of 2012. Spring 2013 I overseeded with shady grass seed mix and white clover which sprouted well. Sometimes watered throughout summer when dry, although some died anyway because it was droughty last summer. Limed and top-dressed again with compost in fall 2013.

This Spring I will overseed again with both grass and clover. As these get established they will be more tolerant of drought and traffic.

Agree the dogs will probably make it difficult to start turf from seed so you could think about mulch or if you want to spend more you could lay out sod.

In my yard, I have the best luck in shade with hostas, rhododendrons, and azaleas. I also have good luck with Dicentra/Bleeding Heart, but I'm sure if your dogs are in and out of the area it's going to get trampled.

The big questioning want to ask - how do you intend to use this space when it's done and planted?

If you want to use it as lawn - a place for the dogs to run, a place for the kids to play - then all the suggestions for perennials and shrubs won't work. Even a ground over like pachysandra or vinca won't work. So terrenes suggestion of adding organic material and a shade grass seed would be your best option.

If you don't want to use it as lawn - then turning it into a big shade garden sound like the way to go. Depending on size, you could add pathways, maybe a garden ws'ing or a bistro set. And fill with shade loving shrubs, perennials and ground covers.

I have the same problem only my lawn is full sun. It's the dogs. I'd love to hear ideas.

I have put down wood chips along the fence where they run, because that becomes a muddy trench otherwise.

I've also put pavers (cheap big ones from HD) in a few areas where they have a "path" like going around the corner to come inside. I need to add more. I'm talking about sticking them right in the grass, not an orderly paved path, but that's another option.

I throw down some seed every spring in the areas that are nicely aerated by the dogs (ok, ripped up) but that's the extent of my effort. I tried fencing off small areas and planting seed one year and it was a huge pain. And the greyhound kept crashing into the fence.

I would love to know of specific kinds of grass that are really hardy! I don't care what it looks like, I just don't like the mud. I've actually started letting the ajuga take over because it is stronger.