Hugs to you! I've have clingy tired, non sleeping babies myself but they've only been singletons. Just out of curiosity have you tried baby wearing? Basically with twins you could put Twin who does not sleep in a carrier on you (I like the Ergo or Manduka styles or woven wraps)
While you tend to happy Twin.
Then after a while (couple of weeks maybe) Twin who does not sleep will hopefully have learned to sleep for longer than a sleep cycle.
It's worked for my babies, mostly who have had reflux & sleeping upright helped them sleep better.

I hope that you are able to R&R soon.