Adding a second Grounding 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.