Q.Last year’s blackberry canes died
Will the new shoots bear fruit? I have some blackberry bushes that were damaged by the severe cold in winter. I had to cut all the canes to the ground since they did not grow leaves. There was significant new growth that should have bore fruit this year. Again, the canes from last year did not grow leaves this spring. I again had to cut all the canes to the ground. I have significant growth from the base and I also have shoots many feet from the main bush. Should I expect the new canes to bare fruit next year or is the bush permanently damaged? Should I start over with the runner shoots?

The new growth this year should produce berries next year. If your bush was permanently damaged you would not be seeing any growth at all.
For more information on blackberry bushes, please visit the following links:
https://www.gardeningknowhow.com/edible/fruits/blackberries/growing-blackberry-bushes.htm
https://www.gardeningknowhow.com/edible/fruits/blackberries/blackberry-pruning.htm