I am not claustrophobic, but I do have a form of social anxiety. Flying first class makes a huge difference for me, simply because I feel like I have more space that is mine. As other posters have noted, there are certain things that may make you feel better. I also always have a water bottle with me, even though I know one is provided in Delta first class. Having the air vent adjusted for maximum air flow toward my head/face also helps me tremendously.

Oh, and gin & tonic works wonders, too.

One caution about the bulkhead seat. I am uncomfortable looking at a wall right in front of my face. I find it easier to look at the back of someone else's seat/top of their head as, again, it gives me the feeling of more space.

I'd do Delta as well since you can reserve the seat you want instead of getting whatever you can find open. Good luck with your flight!!