In your first example it means you are doing the act of moving on your own. The second one can mean the same. I would suggest 'I am moving to a new flat and will live on my own.

In the first phrase, "myself" plays no essential role. If you omit that word, the meaning does not change. It is only there for emphasis. The first phrase becomes more clear if you write it this way: "I myself am moving to a new flat". The purpose of "myself" is to give emphasis to "I". If you omit it, the meaning does not change.