One of the things to remember when you are looking to speak English better, is that English is just one part of language, and language is just one part of communication.
A quick way to build rapport with your listener is to take a technique from NLP and mimic or reflect back their favourite words or expressions. This way, even if you ‘have an accent’ it won’t be as important as the fact that you are connecting with the person on their level and making them feel comfortable by using one of their own favourite words/expressions.
This is also a good thing to remember when you are trying to decide if you should sway your English accent towards British or American. The actual answer is – it depends. It depends where you live or who you work with. I am not suggesting you have 2 distinct ‘fake’ voices, one that sounds like James Bond and another that sounds like George W. I am suggesting that you listen to and imitate the people around you, especially if you are an immigrant. This will help you bridge the gap and increase fluency or at least a sense of fluency until your English improves a bit more.
There is more I could say on this subject but for now the take-away point is to listen to the people around you for their key words and expressions and then repeat them sometimes. Do not be an annoying parrot though. About 3:1 should suffice.