Here are some helpful toweling technique videos intended to help take the fear out of the process. Both are a loose wrap. If you need more control, wrap them like a burrito, tucking the towel around the bird. There are many ways and techniques.
Wahoo:
Tupie:
Holding the towel for a difficult bird to restrain:
You may need to go into the cage with the towel like this. Maybe you have someone hold the bird, covering their head, and you move in calmly and quickly. Think of your approach before you begin. The goal is to do this with the least stress possible to the bird. Ideally you have worked with the bird to be reasonably comfortable with this process and even fully trained the bird to accept toweling. In an emergency situation even the most trained bird might be challenging. The point of this example is to protect your hands from the beak.
