Been there. Done that. Works for a short period of time. A couple of months at the most. Better to freeze than to throw it out. Actually, in my case, throwing it out only means putting it into the compost bin. Still gets used.
Hi and welcome to DC! On storing lemon/orange zest, I simply put it in a small ziplog bag and freeze it. It will lose its flavour somewhat but do add a little more than the recipe calls for when using it.