Our understanding was that

[sumithshah]

Our understanding was that she would continue to cover the costs of school enrollment and room and board for the children, which she verbally agreed to pay with the rest of the family.

which she verbally agreed to pay with the rest of the family.
which she agreed verbally with the rest of the family to pay.
of which she agreed verbally with the rest of the family to pay.
both of which she verbally agreed to pay with the rest of the family.
both of which she agreed to pay with the rest of the family verbally.

OA here is B - however, doesn't which always refer to the noun it follows so then here which would refer to children?

[VP_Jim]

Be careful - the noun (actually, noun phrase) here is "room and board for the children" - not "children".

Tricky stuff!

[sumithshah]

Even if it was room and board for children, then it would not make any sense.

The cost has to be covered as well right?

[VP_Jim]

Not necessarily - the way the sentence is written, it sounds like only the room and board is covered. Another trap: don't make assumptions!

Besides, the two answer choices starting with "both" are way off, so those aren't valid options anyway.