Several years ago I bought some moss killer from Gardens Alive for my mom. I'm not sure what's in it, but it worked very well. They killed the moss, de-thatched, aerated, added a thin layer of compost and re-seeded the grass. Moss still comes back in small areas but the grass is doing better than I expected since the area is somewhat shaded.
Just killing off the moss probably won't fix the problem, unless you also improve the condition for the grass to grow... compacted soil/clay , you might try to work small areas and see what works best...? I agree with
@PhilaGardener , select the best grass seed for your area... and fertilize with enough nitrogen until the grass is established.
My front yard is about half moss, it's shady, compacted soil, I have learned to like it.
There are varied reports on using dish soap, but I may try it on an area where moss is starting in the back, (I think because I trim the grass too low around the flower beds).
I found this...
Ohio State U... although I would test it on established grass to be sure it does not kill your type of grass.