Bring koozies for cans - you won't find any bottles.
As for the tervis glasses I would assume 16 oz would be large enough.
Returning home they provide those cardboard carry containers for your rum (2 bottles per carrier).
I personally always carry my meds in my carry on just so I know they make it, but there is no rule for this - you can do what you are comfortable with - the only thing would be if it was a liquid and then you are subject to the 3oz. rule in your carry on.
Susan & Chris
CTI 2003, 2004
CSS 2008, 2013!