Best wishes to you with navigating this stressful time in your life. This will be a challenge, but you are right that having some FU money will give you more flexibility and options.
Re. your comments on frustration with Craigslist and sourcing furniture inexpensively, I have a few suggestions.
1. There is a button in the Craigslist search refinements box (that panel on the side of the screen to the left of your search results) where you can narrow your search results to only show items listed by the owner rather than a dealer. Right under the header "furniture," there should be a line showing the words "all," "owner," and "dealer." Select the word "owner" and you should be able to filter out a lot of the crappy furniture store ads clogging up your search results.
2. Another easy way to filter out junk posts on Craigslist is to refine your search results by a minimum price greater than $1, since that is what lots of dealers put in their ad to gum up the works when you sort your search results to be from low to high in price.
3. Are you a member of your local Freecycle group? I love Freecycle even more than Craigslist, because it's free (duh), and at least in my area, I find folks to be a bit less flaky.
Finally, I'd encourage you to think hard about what your priorities are and look for those furniture items first. Furnishing a house isn't something you need to do all at once! And trying to do so will likely make things more stressful and expensive for you than if you give things time. Prioritize getting all of your family a decent place to sleep (this could even be a mattress on the floor temporarily), and the rest is honestly just icing on the cake. The last time I moved, I ate sitting on the floor for a month before my schedule/car availability/etc. aligned for me to go buy a dining table. Honestly, it wasn't that bad. Keeping morale up is a more important consideration with kids, and obviously you want to work towards having a "normally" furnished home, whatever that means to you, but it's worth remembering that things like end tables for the living room aren't an emergency.
Rather than thinking in terms of rooms, think through what your top priorities are and focus your energies on sourcing those items items. For example, maybe your shortlist would look something like this: a bed or at least mattress for each person, one kitchen table and enough chairs for everyone to sit at it together to eat and do homework, some basic storage items like hangers or a free-standing garment rack to keep everyone's clothes off the ground, a lamp for each room if there's no ceiling light built in, and maybe a sofa.
I hope that helps. Good luck! You can get through this.