Sounds like creeping beggarweed. It is a very tough one to kill. Roundup might work but keep in mind it is a non-selective herbicide. It is too hot to use most selective herbicides now without causing damage to the turf. Here is a link that might be useful:

That does look similar to Sphagnum moss and if it is that means you have a soil problem. Spraying a poison around is not a cure since all you are doing is treating the symptom of the real problem. Start working on solving this by contacting your local office of the University of Connecticut USDA Cooperative Extension Service about having a good reliable soil test done so you know what your soils pH is (and why it is) and what the nutrient levels of your soil are. You can also dig in with these simple soil tests to see what your soil is like now (most often I find mosses grow on soils lacking adequate levels of organic matter), Structure. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. A good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top. 2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drains too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up. 3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart. 4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer you soil will smell. 5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy. and what is needed to make that soil good and healthy.

I'm not a fan of disturbing the soil unnecessarily either, but digging out unwanted plants so they can't return is a good reason to do so. Cover with mulch/organic matter when finished, the soil layers will repair themselves as soon as possible. An alternative, if logistically possible, would be to cover the whole area under corrugated cardboard, overlapping very generously at any seams, covered with mulch (shredded, not bark) to hold it down and hide it. By the time the cardboard has decomposed, the plants should be dead.

If the stems are velvety I think you do have Rhus typhina, Stags Horn Sumac. It is a spreading, suckering, clump forming shrub and will continue to pop up unless you can remove all the roots and the original parent. But it is native to the US so you might just learn to live with it to a certain extent. It has brilliant Autumn colour.