Ask Mr. Smarty Plants

Good idea!  You already know that the grass is struggling and that moss will grow.

What you probably didn't know is that moss has no vascular system or roots, which means that it takes all the nutrients it requires directly from the air and not the soil.  That means less water (about one percent of what grass requires) and no fertilizers or pesticides.  It is drought tolerant and will come back with a splash of water even after it looks "toasted", making it a very sustainable alternative to grass.

According to David Benner, a retired professor of horticulture who has maintained a moss lawn in Pennsylvania for over 30 years, the trick is the acidity of the soil.  To establish his moss lawn, he covered his lawn with an acidic combination of sulfur powder and aluminum sulfate and waited.  The grass died over the winter and moss blew in on the breeze.  You are already part way there.

You can read a very informative article about Mr. Benner in the NY Times on-line.  You will have to "register" to read the second page, but there is no cost to do that.  According to the article, he deals with leaves by laying a net over the "lawn" which he drags to the compost heap once the leaves have fallen.  You could do that if you find that raking the moss dislodges too much of it (again, no real root system).

There is also a wealth of information on the website of Moss Acres, a company in PA, including an FAQ section on the technical aspects of "planting" moss.

You will need to decide if you are going to take the radical approach Mr. Benner did or gradually change your soil pH (ideal pH for moss is 5.0 to 6.0)and let nature take it's course.