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This will not be necessary. This happens on occasion. When the new growth starts to return, then the old growth will fall, naturally. Now, you can prune off some of the growth to replenish your tree as part of general pruning maintenance. This article will explain how to do this, as this is one of the times that it can be done: https://www.gardeningknowhow.com/ornamental/trees/crepe-myrtle/best-crepe-myrtle-pruning-time-when-to-prune-crepe-myrtle.htm

Your azaleas and crepe myrtle may need more time to recover from the frost attack. If you can cut or scrape a section of lower branch and see a thin layer of green and moisture, then they are not dead. To be on the safe side, take samples of your crape myrtle and azaleas to your local cooperative extension to have them sent for diagnosis.

You will have to let a few branches establish as the new replacement main branches. You can cut everything else off but leave enough to be the next main branches. This article can help: https://www.gardeningknowhow.com/ornamental/trees/crepe-myrtle/pruning-crepe-myrtle-trees.htm

Your area is very diverse climate-wise. This will make it difficult to say whether it will be able to survive the winter in your area. What is the lowest temperature that you receive each year in your area?

Some of your area falls in the lines of the natural climate zone, but some of them are just out of their natural habitat.

Crepe myrtles often leaf out later than other trees, as late as early May. Also, trees that have been exposed to excessive cold may leaf out late, but it should produce leaves by summer. You can check if the tree or individual branches are dead by scratching the bark:
https://www.gardeningknowhow.com/ornamental/trees/tgen/tree-scratch-test.htm

Here are some reasons a crepe myrtle may not bloom:
https://www.gardeningknowhow.com/ornamental/trees/crepe-myrtle/crepe-myrtle-not-blooming.htm

Can you include a photo? Sometimes this can be normal, but depending on the appearance and texture it can be a serious problem whether it be insect, or fungal. A photo will help me to identify the issue.