I travelled back and forth to Australia a lot as a youngster, and probably half a dozen times as an adult and also around the USA living out of hotels for conferences on a few occasions. I would take my insulin (and spares) in a small cool bag with some ice packs (separated from the insulin by a piece of "hi-tech" corrugated cardboard

) and then request ice cubes during the flights (or get it from a bar/free hotel ice dispensers in the US when on the ground) having brought a couple of resealable food bags + extra clip for safety to place it in and stop everything getting wet when it melts.
I'd then place the ice in the food bags into the cool bag to keep it all cool once the cool packs thawed. The cool packs didn't do much at that point aside from acting as a barrier between ice and insulin - the only reason I kept using them was so I could cool them down in a hotel fridge to add more thermal mass to the system if I wasn't able to obtain any ice.
I would also place the entire cool bag in the hotel fridges, so it would all get cold, but if the fridge decided to have a funny moment and get too cold, the cool bag would act in reverse and I would notice what was going on before it was too late.
With the insulin in its box, there is no chance of it freezing even if in direct contact with ice in the bag IMHO (demonstrated by my forgetting to put the high tech cardboard in the right place on some occasions and the insulin still working quite happily). I would still use the piece of cardboard mind you!
I also had my insulin left in an ice bath while on a flight to Australia (it was in a locked metal box which came back full of very cold water at the end of the flight!). Thereafter, I never asked for it to be placed in a fridge on a flight and always kept it with me. Nevertheless, it suffered no ill effects and kept working (bottles of Soluble and Isophane fwiw).
I never needed to source insulin or other medical supplies while abroad, while working in Europe I'd fly back home on occasion having emailed ahead to get the GP to prescribe 3 month's worth of things.
I hope you have fun