No, dormant oil can't be used as a root drench, sadly. The only way that you can get all of the aphids living on the roots is to use a systemic insecticide. I would wait to do this until after blooming is done, to protect bees, and you may want to discard this year's crop. To keep them from coming back next year, use copious amounts of earthworm castings, and compost around the dripline in summer and fall. That will boost the tree's natural defenses to prevent re-infestation.

Never heard of Woolly aphids living on the roots of trees before. Interesting. Just read up on their life cycle.

Yes they go underground and infest the roots. The roots get so weakened that a tree can topple over. A very hard pest to control. If you have any other trees nearby such as elm which are infested and not on your property to control you have a big problem. Since you are in the USA, I would suggest that you contact your state or university agricultural extension agent for advise on control of this pest and for local knowledge of this pest infestation in the area where you live.