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It would probably be a good idea to wait until fall for transplanting, as this is typically the best time for planting crepe myrtle anyway. In fact, transplanting crepe myrtle while dormant would yield the greatest results. The cooler temps will also make the whole process less stressful on the trees. The following article may be helpful to you when avoiding transplant shock: https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

You need to make sure that they have some roots of their own when you move them. Also, you may want to pot the little trees up for at least a whole season before planting out in the yard. This way they will be able to form stronger roots.

I suspect you have a pest infestation, likely aphids. Treat the trees with neem oil (often sold as fruit tree spray in the stores - check the labels). It will treat for both aphids and fungus and is safe for people and pets.

Sounds like you have spider mites. I would recommend treating the tree with neem oil. It is effective against these pests and is a systemic pesticide, so you don't need to reach every nook and cranny the mites might be hiding in. The tree will absorb the neem oil and the whole tree essentially becomes lethal for a short period of time to insects that feed on trees.

If you are certain that the tree is getting enough water, then I would look for pests or fungus. Treating the plant with a pesticide and a fungicide will not hurt it and will take care of these issues if they are the problem.