Deep puddles indicate a very serious drainage issue. The grass will never be right until you get that fixed. You likely will need to cut a swale into the yard leading to a gutter for drainage. Or if that treed area is the lowest spot around, then forget about turf grass and reconsider growing swamp grass. Or you could bring in more soil to fill the low spot. The problem with that is you risk smothering the tree roots and bark eventually killing the trees.

That soil looks sandy in your hand.

Put in some drainage to take care of the standing water first. Consider using some shade tolerant shrubs and abandon the idea of grass. Put down some mulch, pine straw or bark, or other like material and start calling it a planting bed. There are also shade-tolerant ground covers to consider.