In my fantasy world, there's no such thing as a hangover.
More seriously, I think I'd treat the character as being poisoned, with a DC 10 Con save every hour to remove the condition. Strong black coffee, a hearty fry-up, or Irn Bru gives advantage on the save.
After a night of heavy drinking, Con Save (DC set by DM, based on amount of alcohol). Failure causes Poisoned for a number of hours equal to how much the DC was failed by. Dwarves may be cranky, but they can shrug it off
A level of exhaustion seems fine by me but if you find it too severe you could go with a -2 to all ability checks until a long rest or more liquor is taken