If the bumps squish they are scale. Squish every one of them.Spray dormant oil in winter along with the PLC sprays.

Then next yr spray 2-3 times for the leaf curl. Find something better than Dr Earth.

Your tree is badly lacking in vigor. Have you pruned regularly or at all? Peach/nectarine need 25% or more pruned off each yr. Then fertilize enough to get 2ft of new growth each yr.

I never pruned or fertilized my peach trees in France and they were loaded with peaches.I also grew them from peach kernels I saved from peaches I bought.

Thank you for these suggestions! I think I might have too many of these bumps to squish all of them (and also, frankly, they look so gross!) -- do you think power washing with a hose would work? Also, are you able to suggest a brand of spray that you think is better than the one I was using?

I do not prune it regularly but I did prune it last year! By lacking in vigor, do you mean that the leaves are too droopy, or...?

Lacking in vigor because there is no new growth and few leaves in your picture. Peaches/nectarine should look more like this:

Something like malathion will usually kill the scale. In winter treat with dormant oil to smother the over wintering forms. You can't wash them all off. And usually there are the big adults and smaller ones that need killing as well.

Many people use a copper product like Kocide 3000 https://www.keystonepestsolutions.com/kocide-3000-fungicide-4-pounds-387

for PLC. It must be sprayed before bud swell in spring. Spraying later does nothing. If bad spray fall, winter, and early spring.

Lacking in vigor means that it is a bit slow growing. Trees with a lot of vigor grow very fast.

I dont know that power washing is the solution. You dont want the scale or whatever it is to get on other plants. I cant suggest a spray because i dotn spray my tree.