Hey man, I dunno why you were downvoted. A lot of people do the stage name thing for different, interesting reasons. I have a stage name, Micah Green isn't my real name.
I started comedy when I was just about to turn from 19 to 20, so I didn't have a job to worry about, I've only ever worked in restaurants so I've never given a shit about my work finding a set of mine. I settled on a stage name because I have rich grandparents who are very Catholic, and at the time I was afraid that they'd cut me out of the inheritance if one of my cousins on that side saw a youtube video of mine and ratted on me for being a dirtybird. I was worried about my family in general seeing my material. It turned out, in my case, that I was worrying too much. My grandparents aren't including us in the will anyway, and my other grandparents are very chill, so I can say whatever the hell I want. For me it turned out that having a stage name was unnecessary, but the stakes in my situation were much different than the stakes in your situation, you could potentially lose your whole livelihood. It could be that you're worrying to much, but that's more of a gamble for you than it was for me. In the end, I would say that you should do whatever you need to to make yourself more comfortable with just getting the fuck up there, especially if it's something as harmless as changing your name. Just do it.