I understand how helpless you must feel without financial resources to do what you need to do. My stepdaughter has been in a shelter for domestic violence survivors - she went into a double room in late September, leaving her 12 year old with her mother. She's now in a "family room" with a bed for her and a set of bunk beds, so her daughter goes out on weekends. She has applied for subsidized housing and has a good chance at a 2 bedroom duplex for next month. Her situation is different because she wasn't working at all, and she just found some seasonal work (at FedEx) and also some minimum wage work at a dollar store. Because she was a DV survivor and homeless, she qualified for emergency housing and got moved up on some lists.
You are working, and you need to save money for the downpayment, so you probably won't be there as long as she has been. She hates parts of it, of course, but her daughter has also found that the other shelter residents are a like "dozens of aunties" (her words) who give her a lot of attention and give my stepdaughter a bit of a break.
Your situation is not safe, really. You are living with someone who is angry, resentful and volatile. Your 9 year old isn't safe in this home, and that's before we even look at the negative atmosphere with all the swearing and screaming. If she punches one, she can punch others. You can't save enough money if you are helping her with expenses anyway, so it's prolonging your stay there.
Your teen doesn't get to decide. I know he thinks he can, but he cannot. You have to say that this is what we are doing, and we are going to bite the bullet for a bit to work together as a family to get our own place sooner. Sometimes (often) life involves sacrifices, and this is one of those times.
You don't say what your reasons for moving are - and you don't have to - but your kids have already been through some changes, and they can manage some more if they must. It's not ideal of course, but your goal is to get them their own place. I know you thought this friend would offer a better setting, and also that it would be for a lot less time. That didn't work out. She gave you a safety net and you can express your gratitude for that, and you can extricate yourself from her home without condemning her parenting any further. Make it more about how you've burdened her long enough, and you all need to get on your feet faster by saving more money. Be gracious, and be grateful.
The worst thing you can do is let your teen think he's calling the shots. But if you can involve him helping to learn the skills he will need to go out into the world at 18 (budgeting, cooking, doing laundry, making ends meet, making beds....), you can make it more about the positives and his maturing enough to merit more and more freedom. Based on his reaction, he's not there yet. I'm not saying to give him a job of "being the man of the house" at all - he's a teen, not an adult, so he should not have total adult responsibilities. But he can perhaps get a part time job (not sure what the age requirements are where you live) and he can help work to get you all into a situation that will be better for HIM too.
It's not about him being angry with you. That's kind of routine for teens, and you can't go through life trying not to annoy kids! You are showing them strength and determination, resilience and self-sufficiency. This is your gift to them. You have always done things they didn't like, from the day they were born: you changed diapers when they wailed about it, you made them take baths and use car seats, you make them brush teeth and wear seatbelts and eat balanced meals and get enough sleep. They hate all of that until they realize their lives will be much better if they just comply!