Okay so I was able to resolve this issue by upgrading to Yosemite (10.10.2). I had upgraded from 10.9.x to 10.9.y and that did not help the issue, but going to Yosemite did the trick.
When I first ran the program, my character models were invisible. I was able to fix this by following the "solution" from this thread:
http://www.project1999.com/forums/sh...aracter+models
Wine warns that this should only be used for testing and is not really a solution, but it works for me and after playing a little bit I don't see any other issues.