Does sand help with drainage or marshy lawns around a house?
I would not expect that a layer of surface sand will help the problem, you will just have sandier mush.  If you're getting runoff from surrounding properties, more water is coming in than the ground can handle.

If your lot has some grade and the side away from your neighbors is lower, it might help to install a perimeter drain around the lot to capture runoff from your neighbors and redirect it so that it doesn't enter your yard.  

Your comment on the question indicates that there is a lot of surface water that flows over the property, so that suggests that some combination of a perimeter drain and contouring the perimeter of the lot should help.

But there are limits to how effective this can be.  Some of your neighbors' runoff will soak into the ground on the way.  The area soil and grading can create a high water table and your lot, being lower, may be at the water table level when it is high.  I had this situation at a prior house.  If that's the case, a perimeter drain won't totally solve the issue because at least some of the problem water is already below any practical drain system you could install.

Another or supplemental approach is to plant one or more "wet soil" trees in the yard.  These are trees that thrive in very wet soil.  Weeping willow trees are often used for this purpose.  They can soak up water like a sponge and solve the problem of a marshy yard.  This link lists a variety of trees that thrive in wet soil and can be used for this purpose.

Given your description of the amount of surface water flowing over your lot, I would start with a perimeter treatment (contouring and/or drain) to divert the runoff.  See how far that goes in solving the problem.  If the soil is still too wet, consider adding the tree solution.