Hi J., only thing I can suggest that you may not already be doing is for you and your partner to take shifts. That way you can both have a good 5 or 6 hour stretch of uninterrupted sleep. Or alternate nights.
I remember how enormously stressful it is to try to function everyday while so profoundly exhausted. But I PROMISE it won't be forever. Transitions like these take time, different for every kid, especially considering there's a new baby, but he WILL eventually settle down, and eventually everyone will sleep through the night! Really, it'll happen! (well until the NEXT baby, or the NEXT transition, tehehe)
Enjoy them!
:)