What is an easy way to help stop my plants from drying out when I first transplant them?

You can hand water from the top for a couple days until the roots get established into their new soil or hydroponic growing medium. You could also cut strips of capillary matting that go from the top of the growing container to the bottom of the growing container. Capillary matting wicks the water to the top of the growing container allowing the plant to drink from the water supply.