You will need a new name, Poles will pronounce your name "yaddeh" when they read it.
Jade is jadeit in Polish but it does not have any nice connotations in our language.
You can go as Jadźka though ;) (Jadźka is short for (and sort of diminutive but not really affectionate) Jadwiga. An affectionate form is Jadzia (Jadziunia even more affectionate)
if you look for a nickname not a name in Polish you can go for Klejnot/Klejnocik which is a gem or Kryształ(ek) which is cristall.