I am a social worker who works in children’s services. As everyone else is saying, contact your caseworker immediately and tell them everything. Tell them about the alcohol abuse in the home, tell them about the yelling, tell them about the manipulation and saying your the father. Be as honest and thorough as possible and disclose all of the issues with the home. If you tell them all of that and stuff still isn’t being done contact your local children’s advocate. They are literally there to advocate on your behalf.
As everyone else has said, it would be illegal for you to be the father because of your age so if that’s the way your foster parent and her daughter want to go they are just causing trouble for themselves. This 22 year old woman is claiming to have sexually abused you and, true or not, the foster mother is failing to protect you. This should lead to an investigation into the home and (hopefully) that foster home will be shut down. Either way the daughter is claiming to have sexually abused you, you are not going to be required to stay in that home.
As far as being able to afford a lawyer and DNA testing, that is not your responsibility. You are a minor who is in the care of the state. If it gets to that stage (which I seriously doubt it will once the foster family realizes what a world of misery they are opening on themselves) all costs involved will be the responsibility of the state since they are your legal guardian.
Stop discussing the paternity with your foster family. Do not offer DNA testing, do not sign anything, do not engage in any discussion at all. When your foster mother brings it up tell her you are not talking about it and to talk to your caseworker. If she brings you to a doctor or for DNA testing tell the doctor you want your caseworker present. She is digging her own grave. Your caseworker or lawyer may later ask you to do a DNA test to prove it is not your child, but that is something that should be done with them, not with your foster parent. As already said multiple times, there are much larger legal implications in what is going on here and you really need to get CPS involved.