Just be careful you don't get carried away with burying pronouns. You might end up creating a much bigger problem if you end up with a bunch of awkward sentences trying to get around it.
Your sample sounds a bit staccato because a) the sentence structures are similar (He did this. She did this. He did that.) and b) they're the same length. So, one way to kill two birds with one stone might be doing some combining of sentences OR to combine steps within the sentences to a single action (instead of, for instance, walking toward the kitchen and then back out, you could say he stuck his head in the kitchen and then when you move on, we understand his head's not still in the kitchen.).
Sometimes you can cut out a lot of "he did this and then that. Then he did this..." type of stuff by cutting out intermediate steps of actions and trusting your readers to follow. For instance, instead of saying she followed him, if in the next paragraph you started with, "What are you doing?" I asked from the library door., your reader will probably understand she didn't just beam over there. We understand she followed. Here, you seem to have more of a wandering thing going on, so I didn't change that, but I know it's a trap of mine to not always trust my reader to follow me without bringing them through every step. I've been working on that one.
Another thing that might help, depending on how you've written your book, is to intersperse the narrator's thoughts into the action.
For an example of both:
Dave opened his eyes and blinked several times. Oh jeez, he wasn't going to pass out, was he? She didn’t feel like dragging him anywhere. He looked around at first and then walked toward her kitchen, and back out again, then wandered into her living room and library, all the while nodding to himself. What on earth? she thought, trailing after him.