。
I'm dying to drink something cold.
Literal
Cold thing [object-を] want-to-drink [can't-help-でしょうがない].
~てしょうがない ('can't help X-ing, X is unbearable') is parallel to ~てたまらない. しょうがない is a casual contraction of 仕様がない ('there's no way, can't be helped'). Both express intensified, overwhelming feeling.