How do you handle a dog who whines and barks when you leave the room? Could it be separation anxiety?

It could definitely be anxiety. It may be a true case of separation anxiety, or it may be more situational. I watch a puppy for daycare who whines and barks when I go to the bathroom and don't take him with me. For him, he was recently adopted (I started watching him just a few days after his new owners got him), and he's young (about 8 months old when I first met him). He has definitely shown improvement over the month and a half that I've known him, as he has gotten more accustomed to his new family and has spent more time with me and as he has gained confidence that comes with age. Situational anxiety is likely to improve on its own with time and consistency. Separation anxiety requires a lot more dedication and effort. It's the difference between feeling homesick and nervous about a week away at camp and a full-blown panic attack.

I'd take the pup with you when you are able, and give him things to do when you have to leave him alone, like puzzle toys or chews. I would guess his anxiety is just a little nervousness and apprehension about being left alone in a new place, and should improve as he grows more comfortable with you and your home.

It is definitely separation anxiety. I had a dog who was just like that. I could not go anywhere in my house without him following me. It was so bad that his owners would bring him with a bark collar, not because he barked but because he whined when we weren't around him. Finally I referred him to a travel sitter who would stay with him at his home. He seems to be doing better with a sitter in his home although he is still wearing the bark collar.