Ask Mr. Smarty Plants

We are assuming you had the Celtis occidentalis (common hackberry) on your property when you moved  in because most people seem to look down their noses at these trees native to this area, drought resistant and sturdy.  Not only that, but hackberries are among the best food and shelter plants for wildlife; the fruit is relished by birds. From Bug Guide, we found these pictures of Woolly Aphids, not a pretty picture.

You are correct that spraying your tall trees would be a challenge, and the creatures will overwinter in the cracks of the bark on those trees. This University of California Integrated Pest Management website on Hackberry Woolly Aphids mentions several lines of treatment,  but, frankly, we don't think this is something an individual can handle. This article refers to another species of the genus Celtis, Celtis sinensis, Chinese hackberry, but we believe their suggestions for control would still be valid for the common hackberry. 

We hate to tell you this, but we think you may need professional help with the aphids in the trees. You can continue to treat your newly planted native plants, just to help them survive the onslaught. Soon, believe it or not, the leaves will be gone from the hackberries, and the spread of the aphids will stop, for the moment.  As the material we have referred you to says, the eggs of the aphids overwinter in the cracks of bark on the trees, and dormant treatments don't seem to be too effective. None the less, you are going to have to deal with the source, because they will be right back in the Spring.