I am definitely aware of the "Karen" thing. I remember the woman with the off-lead dog in the park who called the police on a black man, the woman who called to report the black family having a barbeque and was later photoshopped into important moments in history,the "Karen and Ken" who recently pointed guns at protesters, there was the woman who prevented someone parking by sitting on the wheel arch, the woman (who may have been joined by a man later) who called the police on a black man who was chalking "black lives matter" on his own house... so I'm well aware of it.
I certainly don't think it's nice for "real" Karen's.
I suspect the OP is in America where first name use is much stronger than here in the UK. Here we tend to offer surnames more than they do.
In the UK, most dealings with strangers will not be on first name terms - we would be Mrs. Smith, Miss White, Ms Green.
In the US, first name use is everywhere.
I think it's unlikely it would make me want to change my name though. I don't believe people make those connections if you are a genuine and ok person. Changing name is fine if people want to. If it's making you feel bad then do it. I don't believe it would be a reason for me.