In general, children are best off with people who love them. I've seen children go from a fairly well-tended foster home back to their chaotic, filthy, overwhelmed house with their father. (Mom was genuinely not safe and removed from the picture.) At their dad's house, they were dirty, exposed to inappropriate movies, and sometimes not eating well. And they were happier. The desperate fear that no one loved them left their eyes. The children might have been dirty, but their confidence blossomed and their responsiveness to discipline improved. Love makes up for a lot.
That said, I can think of many situations under which I would consider leaving my child in the care of another responsible adult for an extended time period. An inability to provide marginally safe housing is one such situation. An inability to provide medical care would be another such situation.
It is ideal to get yourself and your child into a safe situation together. We live in a shared house and it works well. Can you and your daughter both move in with her godparents? Or someone else you could live well with? Check out the book "Radical Homemakers". If you have the skills to be a Radical Homemaker, you could get a "job" in almost any overwhelmed household and save everyone a whole lot of money and stress. Rethink your skills. Find people who need the skills you can offer and can offer safe room and board in return. A "sandwich generation" family (medically frail parents and young children) might be ecstatic to have a trained medical person around to take care of grandma. You are very employable. You might just have to re-think your definition of a job.
If you decide you do need to leave your child with her godparents for a while, consider it an extended childcare arrangement, not an adoption. Or better yet, consider them your new co-parents. You may have to sign a few legal papers so they can make emergency medical decisions and such, but you do not have to sign away all your parental rights. At most, you could grant them *temporary* guardianship while you get settled into a better place.
Think of it this way. Two parents get an amicable divorce. Since the schools are better in Dad's neighborhood, they decide that the kids will be with him every weekday. Mom does stuff with the kids on weekends. Does this make her less of a parent? Of course not! It just means she's making decisions that are in the best interests of her children. She's still present and very, very involved.
Is this traumatic for the kids? Depends on the kid and the definition of trauma. Some kids sail easily through every transition, no matter how large. Some kids get hysterical if you rearrange the furniture. As long as all the adults are working well together for the best interests of the children, most kids manage just fine.
Hopefully you will get that job. But if not, you still have many, many options. You will find something that works for you. It might not be your first choice, but you will find something that works.
Blessings and good luck.
ADDED TO OTHERS ON THIS LIST:
I.'s reaction is both perfectly normal and extremely healthy. It's the reaction of an incredibly devoted mother, a mother who values her child's well-being above her personal feelings. I know people who have put their children in extended care of others for a variety of reasons. It's usually an incredibly painful sacrifice, done only when all other healthy alternatives have been exhausted. The closest possible comparison I can think of is donating a kidney to your child. She is considering ALL options as she ponders what is in her child's best interests....and asking us to help her come up with some more options.
In World War II, many British parents sent their children to be fostered in the US, where they would be safe from the bombing raids. Would you call their reaction an unhealthy abandonment of their kids? I wouldn't. There are times when the most loving action is to evacuate your children....even if you can't go with them. Dying by the side of your children is love. Keeping your children by you so they can die by your side is selfish.
I don't think I. is in that situation yet, and she doesn't think so either. But she's close enough to be able to imagine it and that's scary as hell. If there's only one spot on a lifeboat, I will give it to my child. Obviously, so would I.. As long as there are two spots on the lifeboat, though, it's pretty clear to me that I. will be by her child's side. As would I.