I'm no expert, however my backyard was full of slate, rocks and heavy clay. My husband and I bought about 12 big bags of peat and spread it all over the yard, then spread top soil and then grass seed. It took us about 2 seasons, to sucessfully have soil that wasn't compacted by clay. It is still working, I add leaves to problem areas every fall to aid in the breaking down of the soil to enrich it.

Is the entire back yard a swamp, or is there an area less so, that's still sunny, for tomatoes, etc? The reason for asking, is it's a lot easier to use plants well suited for poor drainage areas, natives especially. Aronias, Ilex verticillata's, Itea's, Clethra's, Swamp magnolia, plus some perennials are well suited. Instead of trying to fight it, work with it.

Here is a link that might be useful: Will grow in poorly drained

I've found that being a successful gardener means acknowledging the basic characteristics of your land. The land is the most fundamental part of a garden. Your plantings should adapt to the local conditions, not the other way around.

You've got a wetlands. Trying to dry it out will be difficult and frustrating and will destroy the fragile amphibian and plant life it contains.

I think the easiest, cheapest, least frustrating, prettiest, and most environmentally sound thing to do is to embrace and respect it for what it is. Let the native plants come back, supplement them with beautiful flowers and grasses that do well in seasonally damp soil, and put your tomatoes in pots on the deck.

The mix you created absorbs water very well. But it could also drain away rather quickly. What's it like there now? You have to watch things over a certain period of time to see what the long term viability of plants will be. Once I put in raised beds alonside this beautiful walkway up to my front steps. Did it during a drought year, actually several drought years. But come to find out, when it did actually rain, my front yard turned into a stream! The bed borders conveniently channeled all the water right up to my front steps! All I'm saying is don't get discouraged. Embrace your yard for what it is, as others have suggested, and make the best of what you've got. I'll bet it's not so bad now. Plus, once you get some vegetation in, it will soak up some of the water. Forget about tomatoes though! Like Carol said, those you can do in pots. But there are other vegetables that might LOVE that soil. Squahes, melons and cukes all come to mind. Blueberries--you can find types in every size imaginable. There are gazillions of other options, edible and otherwise, but I'm just tossing out a few ideas to get started.