I have experience with heavy clay soil as that is mainly what I have been gardening in for many years.
The best methods I found were just consistently working the soil and adding rich organic matter. It is time consuming and vicious work if you do this by hand. I do everything I can by hand. I feel the soil and have a great inner meditation whilst doing it.
If you want fast results I would recommend working in raised beds whilst you work on introducing as much composted organic matter into the ground as possible. Work large areas that you have a plan for but dont do little sections at a time as this tends to defeat the purpose. As soon as the roots of your plants get to the heavy clay again they will not be very happy if they were growing in rich compost and your plants will show you that.
We tend to plant pigeon peas in Puerto Rico because they enrich the planting site and help to break up the heavy clay. That could help for some areas so you could leave them to do their work whilst you focus on another area but I have no idea how they will grow in your area. If planted at the right time you might have some success with them.
Have you spoken to any older growers in your area??? My grandmother taught me HUGE amount about working the clay soil. She is a permaculturist and doesnt even know it. I always find that talking to local growers is always a BIG help. They know lots of the tricks and everywhere I have lived people tend to be pretty nice about giving hints and tips.
I used mainly horse manure from local horses that had been cured in sun and rain for long periods of time. I would basically just break up the clay and mix it in over and over until the consistency is what I wanted. I obtained very good results with this method. You can also add sand I've heard but I never thought that was a good idea.
I am not sure what else I can say. There are many more things you can do to help your soil but I would really recommend starting simple by working in raised beds until you get the remaining planting site ready for cultivation in the ground.
Remember to try to be methodical about everything. I remind you and remind myself. I recently started keeping a diary of the events and of my actions with my plants but it is monumentally hard to have the discipline and time to do this. it does help though when you think hmmmm: how long have these been in the ground? did they come up better in what phase of the moon??
Its not too simple but little by little we all learn and I have learned more from my mess-ups than from my successes. So pay close attention to what you did wrong and then next time you will know; well hopefully.