You have to take action.
I am a single Dad currently in a custody battle for my little 2 year old whose mother also has drinking and emotional problems. Through my efforts, I currently have custody 5 days a week and have since she was three months old.
The drinking and verbal abuse is not only physically dangerous to your daughter, but it will also create emotional scars that will take a lifetime to heal if at all. You can already see the beginnings evidenced by her visits to the school counselor.
Also, it would be a safe bet that there are more problems going on there than you are aware of.
So long as you are a stable parent - (no drugs, alcohol or other skeletons in your closet), you should be able to get immediate custody. Even if you had problems in your past, most judges look at your current situation (e.g. are you recovering and actively doing things to remain in recovery).
A simple report from the guidance counselor on his observations of the home condition will compel the family court to act.
I understand all too well the financial burden a lawyer can create. However, you can go to court "pro-se" or representing yourself. Most judges I have seen are willing to work with people who have no other option the to do that. It shows just how much you care for your child.
Also many courts have a "lawyer of the day". This is someone who volunteers to be at the family court and assist people free of charge. They help you navigate the system, understand the forms, and also provide you strategy and legal advice. Obviously this is not as good as a dedicated attorney but it will provide you enough to be able to help your daughter.
If you do not think you are ready for taking care of your child full time then Social Services should be engaged. It is my personal experience they should only be engaged when all other options fail as they add a layer of complexity.
However if you cannot care for your daughter for whatever reason, you still need to help her get out of that unhealthy and dangerous situation. This is where Social Services can help.
Regarding being 60 miles away, kids can make new friends a lot easier than they can recover from emotional or physical scarring. I would not let the relocation be a factor in your decision.
Anyway that's my advice as someone who is in your position, albeit with a younger child.
Good Luck.
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