Gosh, sorry to hear this. The thing that is important here is that even though the CEO isn't coming to your aid, they are aware. Just do your thing then. Put your head down and do your job and the problem employee that is harassing you will to have reason or back up for their complaints. Until you get another position elsewhere, just smile and act like this doesn't bother you. Be friendly with others in the organization EVEN if they don't return it publicly (because bullies make others shy away for fear they will turn on them). As to anxiety, are you suffering physical symptoms? Would talking to your doctor help you? Overall, you can do things like exercise regularly, splash cold water on your cheeks or forehead if you get panicky, practice breathing. Every time you don't react to their actions, they go down a notch. And you are looking for a job so eventually you will be out of there. And eventually, the ceo may decide that the person doing this a problem and begin to step in more. I am sorry I don't have anything else to offer you but you just do your job to the very best of your ability. Every day. And exit when you can.
I will add that the HR Director is basically a staffing recruiter who I spend part of my time assisting her within her capacity.