The situation is as it is now. I get that some posters feel maybe you have buried your head in the sand a bit previously, but what's done is done. The best thing now is that you keep the solicitors informed of what['s going on re the sale, and try to get that moving forward.
It seems like you've ended up with the shitty end of the stick here, you've basically got sole care of 3 kids, and now have to sell your home too when you are going to struggle to afford a new property. The problem is that there is an order (made 2 years ago now) requiring the house to be sold, so rightly or wrongly that's what you have to try and make happen, even though it may not seem very fair.
I think Winter2020 makes some good points; is there any order around payment of maintenance? You could ask for the arrears to be taken into account when proceeds are split as suggested.
For future payments, if there's no order for maintenance and it's all by agreement, work out if you might get more going via the CSA and make an application, both for your older children and for your child with your Ex partner. Both fathers should be contributing, especially if they don't even see the children any more.
In terms of the guarantor, you might find that some landlords would be willing to accept a larger deposit, or several months rent upfront if you don't have a guarantor (you could use part of your share of sale proceeds for this). See if you can find a property where the landlord is self managing it rather than via an agency, they can often be more amenable. Ask around locally - do you know anyone who rents properties out in your area or knows someone who does? The rental market can be quite competitive so anything you can do to gain an advantage is good.