A couple things
I broadforked our clayish garden soil this year (10'x20'. Broadforking is less damaging to the subsoil organisms. Then I added 40# of gypsum (calcium sulfate, which is water soluble, unlike lime, calcium carbonate) and then added a couple yards of compost on top. The compost was very work rich, so the expectation is that they'll tunnel down into the broken soul.
If you have access to leaves, I'd add them on top after broadforking and let them sit over winter. We them well.
I've just read an article about crushed volcanic rock absorbing atmospheric carbon. On a hunch, I looked up whether it would be good for Clay soil; turns out it is. Yes, it'll cost you, but would be easy to transport in bags in your car.
https://growingorganic.com/soil-guide/volcanic-rock-dust/