"Any person including a law enforcement officer may kill any dog which he sees in the act of pursuing, worrying, or wounding any livestock or poultry or attacking persons, and there shall be no liability on such person in damages or otherwise, for such killing. Any dog that enters any field or enclosure which is owned by or leased by a person producing livestock or poultry, outside of a city, unaccompanied by his owner or his owner's agent, shall constitute a trespass, and the owner shall be liable in damages. Except as provided in this section, it shall be unlawful for any person, other than a law enforcement officer, to kill or injure or attempt to kill or injure any dog which bears a license tag for the current year."
You have the right to kill dogs that are after your animals. Tags or not. Where it says 'except as provided in the section' - read that as 'except as noted above'.
Is it legal to discharge a firearm in your neighborhood?
The law & animal control responses you've received sound almost unbelievable.
My advice-
Document everything. Do your research, know your laws, speak to your neighbor, LEO & Animal Control. Buy a firearm & practice.
So sorry for your loss.
PS - Research small claims court in your state.