I just skimmed through, I didn't read through every post so I don't know if this has been addressed. I think it is great you called Animal Control but I would also contact your local law enforement. They will be able to tell you, for sure, what the law is in you city, county and state.
I had a neighbor's dogs that came and killed a few of my chickens a few months ago. He was talked to, said he'd chain them. He did, for 4 months. Almost a month ago, they were back. This time they killed a hen and a roo. Per other neighbors, this man has had problem dogs that kill livestock for years. I thought it over and decided to call the sheriff's office. A deputy came out for my report. We were told that even though our county doesn't have a leash law, Alabama does and that trumps the county.
He also told me I could have a warrant sworn on the owner for having his dog's at large. I didn't go that far but hoped an "official warning" would make him decide to keep his dogs contained. The deputy was going to make contact with the owner. I asked detailed questions about what my rights were and if the dog had to be doing anything etc for me to shoot it.
A few days later, the neighbor came by. He was upset but I explained that I didn't feel I had a choice as it only took 4 months for them to come back. I told him if they were on our property by the time I got out there, I would have shot them. He let me know that the deputy went out once, nobody home and went out a second time and ended up shooting one of the dogs in the head, didn't kill it but shot it. He said I shouldn't have to worry about that one as it sounds like it's permanently disabled and he's getting rid of or shooting the other himself.