I'm so sorry you're going through this. I've been through a similar breakup, and it is incredibly hard. My DH left for someone else, who he was "friends" with beforehand, including bringing our DD to meet her without telling me before he ended it. It completely broke me at the time. So I understand how devastating this feels for you.
I'll just go off my experience first: it took me about 4months to start feeling better, but it took another 2 after that to start to accept that things had changed. I tried dating in the first few months, but like you I didn't feel ready. I cried a LOT. After a couple of months, he did something which made me realise I didn't realise who he was anymore, and never had. That made it even harder, but also helped me start to make the distance I needed in order to process the change.
Please take the time to cry and whatever else you need to get all the emotion out (without of course turning to alcohol or drugs or anything else destructive as a coping mechanism). Get in touch with your GP or an organisation that can offer you counselling like Relate. Save Samaritans or similar number on your phone, even if you think you won't need it. Don't push yourself to date just because he's shacking up with someone new. Think about writing down how you feel when you feel overwhelmed with it all, and then delete it or throw it away. I used to type out in a blank email or notes then delete it. Focus on your kids, take them out even just to the park or have a floor picnic with a movie, and make some memories together. It will help you and them. They will remember that when they're older, that when dad left, you were there, prioritising them.
Do you have people you can trust to confide in, family or friends?