We have fairly heavy clay soil on our property. I have been actively working to improve it for a number of years.
I would first say to NOT add sand to your soil. In our experience it does nothing to "lighten" the soil. Instead it seems to make soil rather cement-like!! We only did a small area with some left over sand from another project, and it didn't grow anything until I added organic matter and tilled it all up together.
For our clay soil, the addition of the organic matter is the key to "lighter" soil. The pieces of vegatation hold the small particles of clay apart, preventing quite as "sticky" dirt when wet or very dry. The more vegetation, organic material, I can get into the dirt, the easier it is to work, the better the plants do for you.
I would also strongly suggest you get soil test done for proper application of needed minerals. Lime is a great product, but could be you actually need nitrogen or the phosphorus, so buying lime is wasted money.
I am responsible for keeping our fields producing grazing for the animals. Grazing allows us to save a great deal by not having to purchase hay for food. The better the grazing, the more food is available for our animals on our small acreage.
I have the benefit of being able to spread our stall bedding on our fields, so the wood fiber bedding is a huge benefit to the soil as the organic matter. I lightly work the pastures to open the dirt, fertilize, seed any bald spots, drag to cover the seed and get organic matter into the opened dirt. You would not recognize the fields now, if you had seen them when I started. The land was weedy, lots of plantless areas, overgrazed grass areas. The bedding getting worked down into the dirt, helps hold the soil particles apart, provides food for the micro system and helpful creatures in the soil. They actually pull the organic material down and deeply into the soil, to let plant roots use it. Makes it easier for plant roots to grow and spread widely in the looser soil.
So while it sounds kind of weird, you may want to check around for available animal bedding, horse stable with sawdust or wood fiber bedding, to get a supply from. You may be able to get it free for the asking. I would get it before they do composting, or plan to get twice as much if already composted, to mix into my proposed garden location. Compost just doesn't have the bigger pieces of organic matter to hold the soil particles apart unless you really can put in on in HUGE quantities. Straw bedding just breaks down too fast for me, only lasts about 3 months on my field, then gone, unlike the wood bedding. We have a man who brings a small trailer each fall, takes all our horse bedding for probably a month. He gets the trailer and empties it about every other day, brings it back. He is REALLY enthused about how much improved his LARGE garden and fruit bush plantings have gotten with the bedding addition. He had to call on family to come pick, he couldn't use it all for his family. Some bedding he works into the dirt, while other stuff is just used as a mulch around the berry bushes. We clean stalls daily, so the bedding removed is not totally soaked or fithy dirty with manure. His timing is good, saves tearing up the fields spreading during the rainy season in fall.
I also use the wood fiber bedding in mulching my gardens, special trees, and the plantings all are doing quite well. The wood breaks down, goes into the soil, holds moisture for dry times, has lots of worms working in there which makes for lots of air in the soil.
I would probably buy a couple bags of the sawdust or shavings bales at a farm store like TSC, spread that on my garden and work it into the clay if I couldn't get any stall bedding locally. Maybe even a few bags, if my garden was quite large. You want a nice, thin layer of the woody stuff over the whole garden. Not just a sprinkle here and there or it won't be too helpful in getting the clay particles apart. Chopped or shredded leaves in the fall on the garden area will help you too. Worms LOVE shredded leaves, they will break down quicker for you than leaving them whole. Whole leaves will clot up when wet, maples are famous for that, but not a problem if chopped or shredded. Oak leaves just don't soften at all! May blow away if left whole, so mowing or shredding them will keep them where you put them for the many benefits leaves provide. You can run your mower over the leaves to shred or mow them small right into the bagger, then use the leaves for spreading on the garden. LOTS of people drive the streets at night and steal bagged leaves because they don't have enough!! I find it amazing how a pile of leaves bigger than a car, only shred down into MAYBE 3 small bags full. I never have enough shredded leaves. They also make good mulch if you want to use them for that.
Wood organics do use Nitrogen to breakdown slowly over time. Wood does NOT "use up" al the Nitrogen as is sometimes claimed. Wood may lock it up for a while, but the Nitrogen does go back into the soil when the wood is done breaking down, may need that lime application to make the Nitrogen available to the green plants. This was explained by my fertilizer man when I questioned the change in various minerals for application after a new soil test. I find it amazing how minerals interact, all are needed in the correct volume to keep things growing well. My soil needs had completely changed in 3 years, with spreading bedding heavily, working the soil a bit, mowing and grazing the fields. Results were the proof it worked, horses had grazing from May to late Oct., with no need for supplemental hay during that time. And the fields continue to improve, produce needed pasture, with even more animals grazing the same small areas. And the pasture grows well even in summer heat, drought times of over a month in July-August heat. Happy plants with deep roots in aereated soil that lets them get moisture and food. And the flower gardens look pretty good too!
I would sure rather be growing stuff on clay than sand, though the need for the organic material you add are about the same. Both types of soil need a LOT of organic additives to help the dirt. Clay just doesn't dry out as quick. In time, you will be HAPPY with your clay soil. All starts with the soil test, get it done as quick as possible.
The testing service quickly gets overloaded in spring doing farm testing. Waiting will put you behind schedule if weather pulls any surprises on you. I didn't get my fertilizer down last spring because I didn't get my soil test back in time. I test about every 3 years, to make sure there are no surprises taking place. Went from muck dirt to HARD in about 10 days, so I lost my "window of opportunity" with other demands on my time.