For the OP; call your local Animal Control, and the local law, and look up what your state's 'loose dog' and 'livestock ' rules actually are.
Catching a nice dog, and taking it to the shelter, is a good idea. Trying to catch a not so nice dog can get you injured, not best.
While this situation is ongoing, keep your birds in! These dogs won't be your only predators, and your flock needs a safe place to be, when free ranging isn't possible.
File a complaint, call to file another complaint, every time there's an incident. These dog owners are idiots, and need to be 'educated', and owe you for any dead or injured birds, and any other damaged property.
Responsible animal owners might have an escapee very occasionally, and try hard to prevent such incidents, and pay for any damages caused by their critter.
Mary