No, from my reading it's not related to Zinc deficiency.
How many times a day are you putting your hands in water and detergent?
You may be removing all the oils in the skin. Use rubber gloves if this is the case.
Before you use the vaseline, make sure you hydrate your hands by putting them in water. Remove excess water but do NOT allow to dry. Then put the vaseline on. This locks in the moisture.