Putting sand on lawns

My lawn is in a sorry state. After a previous post on here, I concluded it's because it's shady. So I intend to reseed with a shade tolerant seed this spring and see if that helps.
But, it is also quite wet-not waterlogged, but squelchy. So, I'm going to fork holes all over it and put sand in. But, what sort of sand do I need? Is it special stuff just for lawns?

 

I use a medium fine sand.