Hell no. The situation you describe with her drinking is not something I would want to deal with and would not want to put my family through dealing with that. It will affect everyone in your household. You may come home and find her drunk, maybe something broken, maybe she won’t be home and you won’t know if and when she will return. She may be volatile, she may be violent. You will be in a situation where this woman is in your house and you can’t get rid of her. Maybe you might have to get the police involved. Is that something you will want to deal with or will you be like “this is a total fucking nightmare. Why did I get myself in this position?
You said the shelters are full. Is that a fact that you know for sure or is this what she is telling you to be manipulative?
If someone is an alcoholic, and willing to sleep in parks and under bridges in order to continue their addiction, do you want to deal with that?
Is there an end in sight? What will be the remedy for the situation if she comes to live with you? She is probably not employable if she is such a mess. So she would be with you inevitably? Do you have kids? Do you want your kids or your guests to see this mess in your house?
This is the kind of “client” I dealt with while in CPS. We’d send them to treatment, and they still wouldn’t go. They’d give up their kids in order to continue their habit. Alcoholism is a terrible thing but you shouldn’t have to deal with it and your family shouldn’t either, if they don’t have to.
This is not someone who I would want in my house or around my child. I wouldn’t want my child growing up thinking this type of situation is normal. I also wouldn’t want to put my children at risk of harm because of some nut who is in my house getting her drink on.