You have to find the reason why grass fails again and again, instead of repeating the cycle one more time, expecting a different result. Moss points at a situation where something is wrong enough that grass can't compete, and only bottomdwellers like moss can succeed. Check your lawncare practices and do a soil test if the practices are OK.

Just to remind you...NJ is The Garden State. Your grass should grow.

How are you watering after you seed?
How often do you mow and how high/low?
Do you fertilize with anything?

Identify what type grass you have at the present time.
Some types just naturally die and turn brown when conditions arrive to support it.

The conditions you mention....shade....moss, does suggest a damp condition that may exasperate the problem.
So fixing the problem of shade might just correct the condition that invites such browning.

Increased sunlight can be brought about by removal of lower branches of trees that bars it. If this is not possible due to the types of trees (evergreens), then possibly removing the trees might be considered.
In any case, if the loss of sunlight is a fact that helps cause the condition then you have to live with it and find some other way to help the grass grow.
Improving the drainage is another way to help correct the problem. Raising the level or increase the drainage of the area can only help.

Every spring, over lay 1/2" - 1"....of topsoil/compost/triple mix over your lawn with accompanying over seeding of the proper type, can do wonders for lawns.
(do this whether you think it needs it or not)

You don't mention what type of fertilizer you spread....or when you do it....so take that into account.
Early spring....if you haven't done it in the fall....to encourage root growth and spreading of grass to fill in areas affected by winter, is recommended. A high nitrogen lawn type is called for.
Second fertilizing can be done in late spring, and a third spreading, in early summer when temperatures are not so high. (fertilizing should not be done when temperatures are above 80ðF.)
Whether you fertilize in the fall is up to you.....some recommend it---others think its just a ploy by the fertilizer companies to sell more fertilizer.

The fertilizer I speak of is granular---water types, are not recommended for good lawns---they usually run away from their desired place and in the heat can dissipate very rapidly.
A light sprinkling over the fertilized area will help it gain contact with the soil and start its effects quicker.