My heart goes out to you, but no, you shouldn't worry about this; the Bible doesn't teach that we are slaves who can't break away from the wrong things our parents did.
In fact, it teaches the opposite. The Bible tells us that we each make our own decisions about life, and we each are accountable before God for the way we live. The Bible says, "The soul who sins is the one who will die. The son will not share the guilt of the father, nor will the father share the guilt of the son" (Ezekiel 18:20). These are sobering words - but they also should free you from the fear that you'll inevitably follow in the steps of your mother.
At the same time, you would be wise to be on guard and not take even the first step that might lead you down this path. Sometimes the children of alcoholics do become alcoholics (in spite of their hatred for what it did to their parents). One reason may be because they never learned from their parents how to deal with life's problems, and end up following the same path they did of escape and denial. Don't let this happen to you.
Instead, I urge you to put your life into God's hands by asking Jesus Christ to come into your life. Then pray that your mother may find in Him the peace and strength she needs. God loves you both, and life's greatest joy comes from knowing Christ.