Im sorry to say this but you shouldnt have written "fake" on a piece of paper and gave it to your co-worker. That sounds like strange behavior to me- just as their whispering sounds like immature, childish behavior. I know this CRAP is hard to deal with, believe me.
I worked as a promotional liquor model (someone who promotes different brands of beer/liquor) and the women I worked with were usually b*tches who looked down on other people. I was the 'new girl' while the other girls were practically best friends. I could feel them tearing me down with their eyes the first day I started. I felt really uncomfortable and awkward just KNOWING that I wasnt liked. I tried being friendly but was rejected in a very passive aggressive manor. It made me feel like ***** because I thought I finally found a job that I could do that wasnt going to make me miserable. I eventually had to just quit and find something else with a better fitting environment for me.
If this job of yours is stressing you out that much, you need to find something else. There is ALWAYS something else you can do. I prefer to be alone while I work, as do you. Something that worked for me was a 'work from home' job that I found a couple months ago. Its a customer service position at a call center. I had to work IN the office for six weeks and then I was able to log into my laptop from home and take calls from my house. Its working out really well for me since I have bad social anxiety. Try seeking a job like that and let us know if it works for you!

Until then, just stay positive! Your not alone!