I am a native english speaker (born/raised in California) and also very self conscious about the way I sound for different reasons. I suspect many of us have this kind of internal battle, even if it seems relatively insignificant to other people.

FWIW, I think your accent is interesting and enjoyable to hear.

Developing oral skills in language learning out of context (i.e. some place where it is spoken) is a schizophrenic thing. I have somehow put on a british accent two years ago (mother tongue turkish), and the process was like this: watch lots of programmers' talks, and read out loud, being conscious as to my articulation, all the blog posts that I happen to read, and sometimes make monologues alone in my room. And my daily input was vast, as I was having a short two-year break from the uni. Now I'm studying Italian language and literature at the university, and am proving to myself that the way I learned english was spot on, self immersion is incredibly fruitful. One should merely be able to tolerate not understanding everything, written or spoken. And while you definitely have the italian accent, you are speaking is excellent, congratulations.