How do I move out at the age of 16?

Actually it’s funny but a lot of states require you to be living away from your guardians for a period of time before they grant an emancipation, so you’ve obviously got that covered. And no, if they’ve left you in care of people who are taking care of you then it is not abandonment (that’s like saying putting a kid in a year round boarding school is abandonment).

The exact laws and protocols for obtaining emancipation vary from state to state. In most states, the minor must file a petition with the family court in his/her jurisdiction, formally requesting emancipation and citing reasons why it is in his/her best interest to be emancipated. He/she must prove that he/she can support himself/herself financially. Many states require that the minor have been living separate from his/her parent(s)/guardian(s) for a period of time; however, that requires the consent of the parent(s)/guardian(s) in order to not classify simply as “running away”. http://en.wikipedia.org/wiki/Emancipation_of_minors

I would try calling legal aid, this is the texas page http://www.tlsc.org/ or DFPS in Texas http://www.dfps.state.tx.us/Default.asp and find out what your options are

note that an emancipation will only be granted if it is in your best interests… so they are rare…