Hi, I have only read your initial post, it is very late, so I thought I might try to help you rather than not. If you have had all the help you need, then sorry for the repetition.
You say you are a SAHM, so you have been the primary carer of your children. Only in unusual circumstances, none of which you indicate, would a court remove children from their primary carer and grant a Residence (NOT Residency) Order to the other parent.
Your H left the children in your care. If he had serious concerns about your parenting, he would not have left the children in your care. He cannot now allege that your parenting is inadequate and that the children should be removed from you and that he should be granted a Residence order. He left them with you expressing no concerns about your ability to care for them. He didn't take them with him. Had he been worried about their welfare, he would have taken them with him.
Your children, like all children, need one settled home. Flitting about between 2 homes is disrupting for them. They need to know where they live and who their primary carer is. You have always been their primary carer and there is no apparent reason why that should change. Courts believe that change is detrimental for children and should be avoided.
The non resident parent usually has contact (formerly called access) on alternate weekends. Your H has not had staying contact for 3 months. It is most unlikely that a parent who has not even had staying contact would be granted a Residence order. Younger children often find staying overnight away from their primary home, very difficult. If you think they are too young, say so. Children whose parents have recently separated are often very sensitive and do not want further change, ie. sleeping away from home. State what contact you think the children can tolerate and offer only that. Ask your H not to give the children presents as this is confusing for them, it is artificial (it's not Xmas) and looks like an attempt by OW to buy their affection, or an attempt by their father to assuage his guilt at leaving them. State this, it usually stops such behaviour.
If you are a SAHM or if you work part-time, you are obviously more available to care for your children than is your H if he works full-time. An OW is almost invariably never considered a suitable person to care for children if the children's mother is adequate. If it ever came to it, you could always suggest stopping work so that you would be available to care full-time.
Sometimes Fathers attempt to persuade children to come to live with them. Usually this is either to punish an ex W for some perceived fault ( she cheated on him) or, more frequently, to avoid paying maintenance for them or to keep the family home. If your exH starts to put pressure on the children to say they want to live with him, then get expert legal advice.
The OW is getting a man who cheats on his wife. She is getting a liar, a man happy to deceive. He will probably cheat on her too, in time. Besides he will have to pay child maintenance. Not exactly a bargain. You, on the other hand, are free to find an honest faithful loyal partner. She knows that he is capable of infidelity and will have difficulty trusting him. You don't have to live with that.
This is a new beginning for you too. I hope it works out, good luck.