So sorry monnie. I'd ring Centrelink ASAP, because they will backdate your claim to the date you ring. There is paperwork that they will send out for both of you fill in, and the whole process does take a little while. Meanwhile - totally agree, I'd be making plans ie where to live, how to afford it, who to call on for support, where W goes to childcare etc. The more plans you have made, the easier it will be to stay strong and follow through with what you want.
You can absolutely do this alone. You are one tough chick x