SPierce,
Is it possible I could plant some plants in my clay soil, and not kill them?
Yes, indeed. The main thing to watch with clay soil is; Do Not Work It When Too Damp!
If you can work even a little organic matter into that native soil it will most likely grow some good crops.
I move strawberry plants in the spring. I like to move the new ones not the two year old ones. They transplant quite well.
I had a similar thing happen one time. I had some strawberries planted in a container. Next thing I knew, they had jumped out and in two years I had a 12 foot wide patch of strawberries. These were June bearers and the variety sure put out a lot of runners.