How can teachers overcome their hesitation to use drama as a teaching aid in the classroom?
As with most forms of teacher nervousness, the main cure is simply experience. Remember that if you try something in the classroom and it does not work, it is not the end of the universe. You and the students will survive the experience.
If you are considering having students act out parts of plays or stories, or do any sort of in-class presentations, or peer-group activities your main problem is always that there are a couple of either clowns or slackers who disrupt or miss out. To a large degree, making sure all your groups link up to do their preparation on social media means peer pressure will help you (e.g. your intense prelaw/premeds will send messages to the effect: "I am so not going to get an A- because you slackers want to spend your lives flipping burgers!)
Next, have very clear steps for every task you assign students and wander around and keep them on task. Always have a back up plan (an activity you can pull out if the planned one doesn't work).
If you want to have students read aloud, realize that they need to practice passages multiple times aloud to do a decent job of it. Most of my students cannot simply pick up a paragraph and read it aloud in any way that doesn't sound quasi-illiterate (sounding out each word, rather than scanning ahead and shaping sentences as a whole).