I also cut it tall to try and keep the weeds from coming up. Clay soil needs organic matter and lots of it. You may need to just cover the lawn with manure and straw for the whole winter. Your problem is clay soil, shade, tree roots and water. Scott is right.
Organic matter is the answer to most of the problems.
Well finished compost is the best for lawns as it quickly blends in and will disappear from view soon. Leave all your grass clippings even if the grass was heavy and the clippings look unsightly. They won't show for long and your soil really, really needs all the organic matter it can get. I suggest you mow the leaves in the fall and rake them off lightly. Chewed up leaves left to work into the soil will help also.