Catherine I am sorry for your loss.
I have had some of the same situations that you have had and we have been approved to adopt. If they only approved "perfect" people there wouldn't be any foster or adoptive parents.
As long as you disclose everything and accept responsibility for what you could have changed you'll be fine. You can also focus on what you have learned from each of the situations and how it can help you parent the kids. Abuse survivors understand what foster children go through far more so than "normal" people.
If all else fails, don't forget to pray.
