Working with clay can be an uphill battle. Even if you replaced the garden bed contents with sand and mulch, your bed would still be saturated with water after a good rain because they're still surrounded by clay. Dig a hole the size of a shovel blade and fill it with water, see how slowly it drains. That's your garden bed, but larger: a lot of sand and compost in a bathtub full of water.
For the quickest results, build a few raised beds. You'll have to import some clean topsoil (inspect it closely and make sure what you're buying isn't the same stuff you're already dealing with at home). Personally, I would take a quart jar (like a clean mayo jar or mason jar) with a lid, and ask for two cups of soil, please. 8-)
Take it home, mostly fill with water and shake continuously for a few minutes, then set it where it won't be disturbed, like a windowsill. The sand will settle to the bottom almost immediately because it's the heaviest. The silt will take about 5 minutes or more. The clay is the smallest and finest, and it could take a day or two to settle, and some won't settle at all. Anything floating up on the top is organic material like compost. The "ideal" type of soil contains about 40% sand, 40% clay, and 20% silt. (I've never had it myself -- I think it's just a rumor!)
Now you know how to do it, do the same with some of your own soil (take out the rocks and roots and other debris) and compare the results.
If the soil that you're thinking of buying is way off in any direction, it could be a problem -- esp if it looks like the soil you've got in your yard. Unless your 2-cup sample is short on clay... just add some of your own and mix it in. Don't let anyone talk you into something like a big load of spent mushroom compost*, you want real soil, maybe with some compost mixed in. Basically, look at it carefully and make sure see it looks better than what you've got at home.
*Some people swear by mushroom compost. This was usually a mix of straw, livestock manure, sometimes leaves, sometimes stuff like ground corncobs, and gypsum (for calcium). They compost it all together, then steam it and kill all the good micro-organisms, and grow mushrooms on it, usually with chemical fertilizers and sometimes fungicides and sometimes pesticides, and it tends to be high in salts; then they steam it again and sell it. If you want to add some to your beds next year, fine, but just add some to it, don't fill the beds full of mushroom compost.
And get your compost pile going so you'll have good stuff to add to your beds next spring.