Visionary, first, congratulations on focusing on what you need to do. The stress you have been under could LITERALLY make somebody ill. Secondly, the events in your life are unusually pressing. A sick child and losing your home rate right up there with some of the most intense of situations.
First, I would say, if it is possible at all, try to get some emotional support through some kind of counseling. You need somebody who is there JUST FOR YOU, so you can take care of those around you who need you, like your child.
Secondly, perhaps you might consider focusing on doing those things that will make you feel successful, no matter how small the steps . It is so very difficult to feel such loss of control and somehow go forward without thinking about the events.
If part of your program is to take a walk, you can tick that off as one positive thing that you've been able to control that day. I think it is important to do whatever you can to feel that you've achieved some success each day.
Perhaps a short list of daily goals will keep you on track. As you look at it, at the end of the day, you can evaluate it by seeing a row of checks running down your list. This may include wt. loss program items, but it need not be limited to them.
You are right, it IS difficult to be an adult sometimes and being a parent with a sick child makes this so much more stressful. Adding the loss of your home, your shelter, your safe place, makes this doubly hard. Perhaps you can include in your list some fun activities with your child. This will add to the joy of being able to check off successful attempts at staying healthy, focused on positive interactions (not easy when you are in the midst of loss) and cheer up your kid too, regardless of age and medical condition.
Good luck to you.