If something sends DMs in your name, there are at least three possibilities I can think of:
1) An OAuth authenticated app has gone rogue and is sending those DMs.
Resolution: Check your
for suspicious applications having "read, write, and direct messages access"
and revoke access for those.
2) Someone has got your password in some way.
Resolution: Change your password.
3) You are using Twitter via HTTP on an unsecured network, and someone was listening in and grabbed your Twitter cookie.
Resolution: Check the "Always use HTTPS" box on the bottom of the page
https://twitter.com/settings, click "Save", log out, log in again.
If you receive these DMs, make sure that the senders check for these things.