First and foremost, I am so sorry you are experiencing homelessness. Being a teenager applying to college is a big stressor already. I can’t imagine doing it without your feet firmly on the ground.
Second, I agree that your best path forward is FORWARD. Don’t delay college. It is truly the path out of uncertainty. Also, don’t ignore expensive private schools— often they meet need 100% (in contrast to public universities). Ask for an hour timeslot with your high school college counselor and come up with a list of schools that meet need. Apply to those.
Third, stay where you are until your Mom really is settled— housed, employed, all of it. Do your part around the house you’re in— dishes, laundry, anything you can. Show them you appreciate what they’re doing for you. This will go a long way towards making sure you are welcome to stay as long as you need to.
Lastly, I wish I could wrap my arms around you. Please keep us posted.