From your post, Sunfield, I gathered that you bought the necklace knowing it was not silver, but not knowing what it actually is made of.
I would ring and ask if there is a possiblity of a refund if you return it, as you have bought it, unaware of the fact that it apparently contains something you are allergic to, as you have never before had an allergic reaction.
I doubt you will get your money back, but it is worth a try.
I would advise against either coating it with nail varnish and simply dispose of it. I didn't think it was still legal to sell anything containing nickel without clearly stating the fact, as many people are allergic to nickel.
Whatever the substance is that you are reacting to, the reactions are likely to become more serious if you go on using whatever causes them.
If there is no description of what the metal the necklace is made of, ask the store if they can check this for you, so you can avoid the allergen in future.