I'm almost certain that if you appeal you would win, but firstly it would be better to submit additional evidence to prove that you have "sole responsibility" for the children and their upbringing and additional information with regard to the childrens biological father, including cotact information, then ask them to re-consider their decision. I don't know exactly how you go about asking for a re-consideration, but I think there may be others on this forum who may be able to advise you on this.
If you can't get your case re-considered, then the next step would be an appeal. If you have to lodge an appeal then I believe you have to do it within 21 days and if you appeal directly to the British embassy in Manila, they will automatically look at your case again before they send it to the UK for a full hearing.
I get the impression that the ECO's are a pretty cynical bunch and when they look at your case again, they will make a final decision based almost entirely on whether you are likely to win an appeal or not.
Iain.