My current electrical system appears to have the ground and the neutral wires on the same bus bar. If I had a second grounding bar, do I need to move ALL of my ground wires to the new grounding bar, which in turn would free up space on the ground/neutral bar?
Your neutral and ground bars must be separated, UNLESS this panel is the main service entrance panel, meaning this is where the line comes in from the utility meter and the neutral is bonded to the grounding electrode conductor (the wire going to the ground rod). In that case, the grounds and neutrals can be mixed on the same bar.
The problem I see is that if you have run out of spaces for both the grounds and neutrals in that panel, there are likely other problems with your system because you should not be able to run out of spaces unless something was done wrong. For example you may have wires doubled up on breakers where they should not have been. Many breakers are NOT rated for more than one wire per lug, those that are tend to come in panels with larger ground and neutral bars. If not that, then someone has installed too many "twin" (1/2 space) breakers to increase the number of circuits beyond what the panel was designed for.
Since your panel is a main service panel, neutral and ground must have an equipotential bond inside that panel. As such, it's under an exception that allows all neutrals and grounds to be just spammed together on either a ground bar, or a neutral bar that is grounded.
If you add a ground bar, you can place any neutrals and grounds on it that you please.
However, elegance calls for putting all neutrals on a neutral bar, and all grounds on a ground bar, and then assuring the neutral-ground equipotential bond is intact. I for one don't believe "green bar-grounding screws" are worth the paper they're printed on, because I've seen ground faults burn the threads off them and they fall out. I prefer to have one or more heavy wires connecting the neutral to ground bar, and this provides a side benefit: if you suspect ground-fault leakage, you can put a clamp ammeter around the wire and know for sure.