Oh that is awful. I think the solution has to be to correct the drainage. A contractor or landscape architect might be who to contact. I don't think plantings will really cure that. But there may be a way to divert the water and you could create a rain garden or pond.

French drain, pond installation, or reeds. Maybe a combination of all three. You could install a gravel pond bed that would be dry or wet depending on the time of year. Is there a house next door back there that you have to deal with or are you on the edge of the woods or open space. That will affect what you choose to do.

I'm on the edge of woods which are 10x more swampy

I'd check with a nursery and see what plants are compatible in water for your zone. We use papyrus, water grasses, and reeds here in California. And I have papyrus in soil along the house. I have reeds in wet spots. Lookup wetland boardwalks for ideas to create a crossing for the muck and side plant it out with water plants to make a little ecosystem back there. Obviously that's not cheap, but perhaps you could use remnant materials or have a school design crew do it as a volunteer project if that's available in your area. You cannot exactly fix the swamp in the woods but maybe you can do a little something in your spot to reclaim your yard.