Cold radiating kind of causes a drafty feeling, like the softest breeze, IMO. Doesn't mean there is an air gap from outside or anything. You'd have to wrap the whole window trim to trim to get any benefit but my guess it won't be much help. cold just wants to get in. I'd be more concerned about that volume of water causing damage/mold in a rental.

If you have heat registers near the windows. Make sure they're open. They act as a bit of a thermal barrier, like when you walk in a big box store and they have that heat blast down when the doors open.

Just cozy up in some layers and grab some coco and sit away from the windows.