Because I live and work in a very diverse area of the country, some of my clients are from other parts of world and speak with an accent. Many will ask me about accent reduction classes. I say – being understood and speaking clearly is far more important than reducing an accent.
One sure way to accomplish precise speech is to practice enunciating while reading a book or newspaper article. Unclench your jaw and open your mouth more as you pronounce all of the vowels sounds. Don’t be afraid of exaggerating the movement as you practice at home. (Most likely it will feel more odd than look odd.)
It is my experience that the mere exercise of opening your mouth fully – to form every word – will enhance your ability to be understood.