First, I would ask the children's teachers to send homework, which is clearly within your children's ability range and matched to their preferred learning style. If they are working on material that is too hard, they may balk because they don't want to make mistakes in front of their parents. This one is just the same, and the teacher told me you really understand this. If the kids are working on specific IEP goals, the teachers might provide you with a packet of materials and activities that would let them practice skills that have been taught at school that day. Also, think about fun ways to reinforce new skills that don't feel to the kids like homework.
Some families get more done at homework time when they set up an incentive program.
In terms of the independence, they may still need you to help them get through the work. If they do the same kind of work independently at school they may just be dragging their feet to keep you by their sides. You can build this independence into the incentive plan. You've just done this one with me. Then next one is just like it.