We have some dead branches in the interior of our rosemary bushes and would like to rejuvenate them. Should we prune them really hard and cut out the dead stuff? Suggestions?
Yes, you can certainly prune any dead branches.
It's difficult to know what caused them to turn brown without actually seeing your rosemary bush, but, most of the time, the cause is poor soil drainage and inadequate sunlight.
Right here on GardeningKnowHow there is an article on rosemary: https://www.gardeningknowhow.com/edible/herbs/rosemary/brown-rosemary-plants.htm