My solution to the sticky petroleum jelly cotton balls is cling film or whatever the rolls of plastic film you use on leftover food in the fridge is called (the thin plastic will stick ("cling") to most dry surfaces, included itself). Just roll the PJ cotton balls in cling film, then stick the roll into whatever container you're using.
In really high temperatures I would expect some petroleum jelly to become liquid and make its way out of the cling film roll, but really high temperatures is not a problem where I live.