I used to have EXACTLY this "problem" for almost 10 months. I live in a town with only 28,000 people.
You may not want to hear this, and I kept denying it for a long time...but the truth is, you have a fear of rejection. I was afraid - and you're afraid - that if you get rejected by a girl, everyone will know. After wallowing in this fear for 10 long months, I've finally decided WHO CARES?!
Who cares if everyone in the world knows you got rejected by some dumb broad. It won't matter, it's a part of life. No one goes through life without rejection. When you apply for a job, to a school, or pretty much anything in your daily life that happens could involve REJECTION. Apparently we've made rejection from a chick into a big deal. It shouldn't be. Yes, I'm emphasizing these things half because I'm just getting over that hump of realizing these truths myself.
Point being, if people talk about me approaching chicks, at least they know I'm a bold guy who's not afraid to go for what he wants. Let them talk...but at the end of the day, it's just talk. Guys will talk about you. They'll probably respect you for having balls or hate you for it. Other girls may think you're a player. Lots of rumors could spread...you're this, you're that...etc.
But I've come to the conclusion that I'm not going to live my life giving a **** what people think any more. No matter what I do, some people will not like me, straight up. I'd rather be on the field living my life than sitting on the sidelines and worrying about how people perceive me, etc.