Syringes are allowed through security, per the TSA, as long as each syringe is printed with the prescription information, and the names match your ticket. It also appears to help to have a short note from your doctor explaining why you need the medication.
I'm not sure how cold your medication needs to be, but something like this might work:
At least for the short term. If you have the option, book a "suite-type" hotel room (Doubletree or Embassy Suites are the two that come to mind), as they'll always have refrigerators - and a LOT more room for you if you're gong to be somewhere for a few days.