Help, I'm an extrovert trapped in an introvert's mind?

I must come off pretty rude because, for the majority of the time, whenever I see someone I know I want to say "hi" and have a short conversation with them but my anxiety goes up and I end up ignoring them. I always promise myself that the next time I see that person I'll talk to them, but in the heat of the moment I just can't. I know what to say -- it's all in my head; I play it over and over -- however in the heat of the moment, my body gets tense and my brain and mouth just shuts down. I never start conversations, but when others talk to me first I'm always alright: smile, make eye contact, strong voice because I know that they want to talk to me. Usually though, the conversation does not last too long before I start acting like my old nervous self again. And the next time I see them I know I should talk to them but, again, my anxiety goes through the roof. It's just very hard to keep that continuous stream of confidence and sometimes I feel myself getting better, but other times I feel so isolated. In the end, the trend is waxing and waning downward and becoming more difficult over time. It's frustrating, because I want to make friends and have fun but it's really difficult to just let go of my expectations and be free and spontaneous all the time. People must think that I have something against them, or that I just want to be alone but that is definitely not true. But that is what I feel everyone, even my parents, feel of me.