I've used a pitchfork to expand my garden beds. Same concept, but a broad fork would be much easier on the back. Both would allow you turn the soil fairly deep 12-15" easily. You may even want to use a long metal digger.
Where ever you are trying to loosen the soil so that water penetrates to the roots - come back after loosening with the broad fork with a tiller and turn in some clean mulch/saw dust, compost, coffee grounds, topsoil, etc. (This is where some will say to look into hugelkulture.) Don't add sand. It actually fills in the air pockets you're trying to create. The goal is to add life to the soil so the clay isn't so heavy. And life will create a space (air) for itself (roots, digging, decomposing...). You might be able to plant a cover crop for the winter, something like buckwheat or alfalfa to get some roots going into the soil. Then chop/drop next year and lightly turn/rake it. If you don't have time to do that, don't sweat it and let the dandelions grow. And next year throw a bunch of beans in that plot. They root fairly deep and are productive.
Search some of Endurance's posts. I'm pretty sure his garden was on a significant slope where he did some terracing, swales and other water flow management.
