Moss killers really do work, but if you don't get a good even coverage you'll get dead black patches which you'll rake out...and then be left with bare mud in which dandelions etc will grow.
If you want a bowling green - dig it up and start again. If you want a better lawn, start with hiring a lawn scarifier to rake out all the thatch, moss etc and make sure you re-seed the bare areas at the appropriate time of year (not my expertise!)
As Tail End Charlie says - unless you tackle the underlying problem (too shady, poor drainage?) you're onto a losing battle with any solution.