The 'specks' are likely scale, as well as what you see on the branches. Euonymus scale can be very damaging. If the infestation is very serious, it is sometimes best to cut your losses and prune the plant all of the way to the ground or to remove it entirely. Euonymous will recover from that kind of pruning, in most cases.

Horticultural oil is often recommended for the control of scale insects, but this is not the time of year that those products can be safely used (heat).

Here is a link that might be useful: Images of euonymus scale

I kind of figured I'd have to cut them way back. Is there anything else I can treat them with since the hort. oil isn't recommended in the heat? Will soapy water do any good?

Check with your Extension for crawler dates in your area, but I'm in zone 7 and know we have already had the second generation a while ago. Sevin, oil, soap, Malathion, and permethrin are recommended by VA Tech to control the scale in the crawler stage.