You can appeal again. But because the date of your last day isn't exactly right, or that the employer "lied under oath" aren't going to be biggies at all. As I have said repeatedly, they work in these agency hearings under the assumption that one or both of the parties may be lying. They go with whom they think is "most believable."
It sounds an awful lot like you didn't show up, because you assumed you were fired, and by not showing up, you assumed your way into what looks like quitting territory to the unemployment system. They rarely change these decisions in the board of review appeal. You definitely need to be looking for other jobs and moving on. As M&G as pointed out, this should be a learning experience. And spending ANY time during your interviews for next jobs talking about this job will not be at all productive.

