Она оставила его разбираться с последствиями.
Она оставила его справляться с последствиями.
I wonder if "справляться" is about getting the entire process of dealing with a situation over with, whereas "разбираться" is more detail oriented, focused on how you sort out a more complicated situation.
"Разбираться" is indeed may be more detail oriented. But the real nuance here is the likelihood of success. For "разбираться", the problem is assumed to be solvable, even if one can fail to get to the bottom of it. For "справляться", it is implied that one can totally fail at the task.
So, "Она оставила его разбираться с последствиями" - means that one was left with a task of dealing with some consequences - something tedious, but doable. "Она оставила его справляться с последствиями" - means that one was left struggling with some consequences - meaning that the task can be overwhelming and even lead to some kind of harm to the one who's doing it.
In the context of that brief example there's no perceivable distinction. The meaning of "разбираться" here is "to look into (smth.), to process (smth.)" ; for "справляться" it's "to (begin to) cope, manage, deal (with smth.)".
