First you need to understand that buttercup loves moist, acid soil so unless you change those conditions the weed will return even if you use chemicals. To get rid of the dampness raise the level of your lawn with a one inch layer of gravel, then 3 inches of  topsoil so that the buttercup and old lawn is hidden under 3 to 4 four inches of soil for a fresh start at a new lawn. All this is a very expensive and time consuming project however. You may want to just slowly change the acidity of the soil be adding Super Sweet lime every year, spreading sand on top of the soil and aerating every spring to encourage better drainage. Keeping the lawn mowed so the buttercup cannot bloom also helps. In a few years you will see less buttercup but still there will be buttercup in moist, acid soil. On a positive note, you can dig this spreading weed out of the lawn with a screwdriver to uproot the mother plant and then follow the tendrils of the young plants as they spread through the lawn. Hand weeding can be rather pleasant on a mild day with the birds singing and a bucket filling with weeds. At least that is what I tell myself as I uproot buttercups from my own shaded, but small lawn,  At one time I had a larger lawn but then replaced the buttercup and grass with lamium and vinca minor groundcover that crowds out the buttercup. Keep growing, Marianne Binetti