Yikes. This topic can get a bit messy, because I've heard people on all sides argue about the Priest wearing his wedding band. The positions I've encountered are:
(a) Wedding band yes, even during Liturgy.
(b) Wedding band yes, but not during Liturgy.
(c) Wedding band no, and especially not during Liturgy.
I happen to support (a) personally. The priests of the GOA generally fall into either (a) or (b), and the difference is probably generational (i.e. it depends on what influences they had from their hometown priests and, especially, seminary professors and bishops).
Would I wear an additional ring? It depends on what's on it, but probably not. I wouldn't wear any type of jewlery (other rings, watch, wristband/etc.) save my wedding band (and the cross around my neck) at Liturgy, though.