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The general advice for pruning most established flowering shrubs is to prune them after they have flowered and to prune up to one third of their current size. Generally speaking, established flowering shrubs can usually recover if you don't prune them too much.

But you don't have to necessarily wait until after the plant has flowered. If you want to prune the Indian Hawthornes now (in the spring), you can safely do that. Just don't get too crazy with the clippers or pruning shears and cut them too far back. Cutting 1/3 of their current size away should be about right.

Of course, once you prune them back, you will not be able to enjoy any flowers and berries this year. However, they should bloom again next year.

They are not more prone to transplant shock, but any plant can be affected by transplant shock. Plus, the older a plant is, the more prone it becomes to transplant shock. Since you moved them twice in a year, this would make the problem worse. The best thing to do is to increase watering and give them some phosphorous, if you can. A tablespoon of sugar in a gallon of water once a week until they start to recover will also help.

Here are some links and articles to help you determine if treatment or removal of the shrubs is the best choice.

http://www.clemson.edu/extension/hgic/plants/landscape/shrubs/hgic1078.html
https://www.gardeningknowhow.com/ornamental/shrubs/indian-hawthorn/growing-indian-hawthorn.htm
https://www.gardeningknowhow.com/ornamental/shrubs/indian-hawthorn/indian-hawthorn-pruning.htm

Indian Hawthorne transplant best in spring.

https://www.gardeningknowhow.com/ornamental/shrubs/indian-hawthorn/growing-indian-hawthorn.htm

Fall and winter are the best times to transplant shrubs in Texas, so you can transplant yours in January. Gardeners in some regions may need to wait until early spring if the ground is frozen in winter.

Transplanting Hawthorn shrubs should take place in spring and they are happy in any well draining soil, full sun location.

It is best to root prune established shrubs 3 months ahead of time. Root pruning is done by taking a sharp shovel and digging down 8 inches down and in a diameter at least 24 inches around the base of the plant. This cuts off the long roots that won't travel with the plant and encourage small roots to form.