I am very sorry to hear this. These are brief answers to your questions.
"From what I have read, his portion of our house and all his belongings will go to the children,"
Yes, however children under the age of 18 cannot inherit until they are 18.
"and I will not be able to apply to be an administrator of his estate,"
Since you were not married then it is the children who will administer the estate.
If any of the children were over the age of 18 then they would apply themselves to administer the estate.
However, in the case of children under 18 then the person with parental responsibility for the child/ren is entitled to take out the grant of administration on their behalf. So, you, as the mother of the children under 18 can apply on their behalf.
"but may be able to be a trustee of the children's trust where his assets will go. "
Yes, you will be one trustee. However, where there are children under 18 there must always be at least two trustees. So, once you have the letters of administration, then you will need to appoint a second trustee as well as yourself.
There is specific wording that you need to use to appoint a second trustee so I suggest that you seek specific advice about that from a professional when the time comes.
The trust capital will need to be invested until the children turn 18. However, any income the trust receives (eg interest or dividends etc) can be used to help the education etc of the children.