First off, sorry to hear about that mate. Hope you are both OK.
As you stated the 'not rushing in to things' part is the most important. It appears that both of you know you like each other. The problem with moving in together because of what has happened may or may not cause issues for you both. Especially as there are her kids to think of.
If it was me I would not rush in to moving in together, if she already has an apartment you could move in for a period of time whilst looking for a new place. If that works well and you are all happy then decide if you either stay or you look for somewhere together. It takes so long to get used to habits etc of partners when you start off living together, you could find out some you do not want too
