Did this at the start of this year, actually.
Fortunately it was in an amicable situation and there was no massive rush to move.
AirBnB might be an initial option, I suppose, but I'd be wary of the cost; I'd also be super open about what was happening there - make sure the owner knows that you and the children are moving out of the family home and just looking for a temporary place while you sort yourselves out.
Best case scenario, you get a discount and they want you to stay long term and it all works out. Not massively likely, but you never know.
I think better just to search the local rental market. There'll be furnished places ofc, but its a competitive and expensive market out there, so be aware it'll take time, but will ultimately be worth it.
I'd perhaps just internally dampen expectations of a speedy move out and just calmly start looking...best case scenario is that you see a place in the next 2/3 weeks that you can move into, get the admin sorted (and cash ofc) and you're in within a few weeks.
But that would the minimum time I'd expect.
In my experience, I started looking in August but didn't move in anywhere til late Jan.
Missed out on places due to competition, a little bit of prevarication and hesitancy and generally just adapting to a new and strange situation.
Unfurnished could work too tbh. Mine was largely furnished by a very few things I took, but a lot from charity shops. Stuff I got just to make the place seem like a home are probably permanent fixtures now.
I got lots of plants too.
Don't forget, if you're ordering things, you'll need to get your ducks in a row as there are long lead times for beds/sofas etc.
Could work out well if you order in Jan when the prices are lower.
But it all takes much more time than you might think...but it will come around and you'll be in a refreshingly new place both mentally and physically by next spring - just stick with it, plan and prep.