Not storing or even coming into contact with passengers’ medication was a point of emphasis during this year’s FA recurrent training.
Passengers are allowed to bring bring 5.5lbs of dry ice on board. I’d suggest doing that, and packing accordingly.
There are, however, sharps disposal containers available. Ask an FA for one should you need it -- and please don't throw it away in the lav or other trash bins.
For what it's worth, generally, widebody aircraft do have chillers for the meal carts onboard, though the carts are generally kept shut until it is time to use the meals -- policy dictates you will not be allowed to store anything in these. Once all meals have been used, the chillers are turned off (otherwise, the galleys become chillers too). There are no freezers to re-freeze ice packs, however.