As I have posted earlier, most people who are working on reducing their accents should also constantly be trying to improve their vocabulary. It’s particularly important to focus on improving your understanding of and ability to use idioms, expressions and slang (when appropriate). Using this type of speech will give the impression that you have native-like experience with the language. This has a psychological effect on the listeners, making them “hear” a less strong accent.
Last week I was teaching a student who is an an actress here in LA. She was was working on a TV script that contained a lot of idiomatic expressions that she didn’t understand. I told her I would look for a good book to help her and I found a good one that I would like to recommend to you. I just received in the mail today and I like it a lot. It’s called “Dictionary of American Idioms and Phrasal Verbs.” I got it on Amazon for a pretty low price.
It’s a lot thicker than I expected. Just be careful, a few of the expressions are a bit old fashioned. You might hear them in old movies but young people don’t use them today. However, most of them are very common and you should try to learn them. I think this will give you more confidence when speaking with native speakers.