You can improve the drainage! I wrote this for someone else, but I think you'll find it applicable also. I've done this many times and seen this work to drain the clay in OH, in the subsoil compacted with heavy equipment after a housing development is built, and here in AL, it helps make the too-well-drained sandy soil hold moisture longer, have more tilth and fertility.

If you've got more time than energy, like I do, smothering and lasagna is the easiest way to start a new garden bed for free, or almost. Sooo much easier than digging up grass. Just spread newspaper (about 10 sheets thick) or cardboard, overlapping well, until the area you want to be a bed is covered. Then cover the paper with 4-6" of finely shredded mulch and wait for the grass to die, usually 4-6 weeks but could be longer for some grasses. I've done this many, many times.

My latest one is really ugly but I'm just trying to make lemonade out of lemons with this one... drought (probably aided by grubs) killed the grass here so I decided that would be the new sunny front bed I was considering. I did dig out a little spot that had hardly any grass and put some Cannas and Gladiolus there, a tiny baby maple tree, some Hibiscus cuttings which still just look stupid 'cuz they're "dead" sticks in the ground, then kind of working around it with smothering, and bark chips, which aren't my preference but I had them available. They don't stay in place if it ever rains really hard. Anyway, with this, I'm not planning to leave the bark chips there, they're just making sure the newspaper is held firmly to the contours of the ground to block the air and light from reaching the grass, which is what is needed to kill it. Whenever I can find more shredded hardwood, I'll replace the bark chips.

Anyway, the newspaper decomposes and does not need to be removed later, just dig through it to add plants in the ground.

I've also smothered grass with stuff that was handy, but does have to be removed to use the bed, like sheets of metal, old egg crate mattress topper, the bags of mulch that will cover the spot, whatever's handy. I think it's easier to wait for the grass to die than dig it up, and I don't mind if it has to get more ugly in the process of getting more pretty.

One other benefit of smothering with a leave-in-place substance like paper or cardboard is that the weed seeds that may be in the ground are unable to germinate as they might be if you just dug up the grass and/or tilled.

The lasagna comes into play if you add amendment layers to your smothering. For example, you could put the paper/cardboard, then kitchen scraps, ready to use compost, leaves, yard trimmings, whatever organic material (OM) that is handy, then the mulch (or not, if the other stuff is a thick enough layer to hold the paper in place and block the light.) It's not necessary to have lasagna layers when smothering, but when planting later, there's a huge improvement if a lot of
OM was placed...