Killer thought Cross liked Dream.
Cross thought Killer liked Nightmare.
Killer and Cross had the great very original idea to pretend to be them for the other to fall in love with them.
They both get suspicious but dismiss it every time.
until a slip-up reveals the whole mess of a truth.
The real Dream and Nightmare are just done with them, leaving them on their own while they go on vacation.
(One or both of them (Killer and Cross) can try to tell the other intentionally but doesn't in the end, then the slip-up happens, if you want more angst :>)
(Also Dream and Nightmare doesn't care about identity theft anymore, they're just done with everything so they just want to chill somewhere else xD)