Ok, to get through a day at this point with no crying is a HUGE thing. Well done!
Re: the solicitor, £750 is not a lot for an initial payment (my divorce/financial settlement ended up costing me £15,000 and the ex £25,000 - and then the judge ordered a costs hearing because the ex had dragged his feet for so long and costs were awarded against him. So.....it cost him £40,000 for being an arse about the whole thing - on top of what he had to pay me as a settlement). What you need to do is speak to solicitor about applying for legal aid now (which you should be entitled to). Legal aid will then pay the costs so that you don't have to find the money, but you will be expected to repay it out of any settlement you get, unless you can get costs awarded against the ex.
Please don't even consider taking him back for the sake of the children. To my absolute regret the first time my ex deserted us and whored around for 6 months I filed for divorce and then he put so much pressure on me (with 3 kids under 4) about wanting his 'family' back, and knowing what a 'huge mistake' he'd made that I felt horribly guilty for the kids and took him back, cancelling the divorce.
All that happened was that he was a complete bastard and the marriage limped on for another 3 years until he (once again) announced he didn't want me and didn't think he was a family man any longer. I then had to start the whole process again, which was much harder on the children, who were then 3 years older and missed him more.
Keep strong. Keep going. You're doing brilliantly. 