Disclaimer: although I am a lawyer, this is somehow not legal advice
The Texas animal cruelty statute defines animal cruelty as:
§ 42.092. Cruelty to Nonlivestock Animals
(b) A person commits an offense if the person intentionally, knowingly, or recklessly:
(1) tortures an animal or in a cruel manner kills or causes serious bodily injury to an animal;
(2) without the owner's effective consent, kills, administers poison to, or causes serious bodily injury to an animal;
(3) fails unreasonably to provide necessary food, water, care, or shelter for an animal in the person's custody;
(4) abandons unreasonably an animal in the person's custody;
(5) transports or confines an animal in a cruel manner;
(6) without the owner's effective consent, causes bodily injury to an animal;
(7) causes one animal to fight with another animal, if either animal is not a dog;
(8) uses a live animal as a lure in dog race training or in dog coursing on a racetrack; or
(9) seriously overworks an animal.
Therefore, shooting paintballs or air gun pellets at or killing a dog attacking livestock would be animal cruelty under (b)(2) and (b)(6) if not for the following sub-section:
(e) It is a defense to prosecution under Subsection (b)(2) or (6) that:
(1) the animal was discovered on the person's property
in the act of or after injuring or killing the person's livestock animals or damaging the person's crops and that the person
killed or injured the animal at the time of this discovery; or
(2) the person killed or injured the animal within the scope of the person's employment as a public servant or in furtherance of activities or operations associated with electricity transmission or distribution, electricity generation or operations associated with the generation of electricity, or natural gas delivery.
In short, in Texas, you can injure a dog with non-lethal means during or after it attacks your chickens. You can also kill it during or after the act. The animal cruelty statutes of many other states are similar. In fact, most of the states' whose animal cruelty statutes I've examined are similar.
However, the best options are probably:
1) If you want to remain "on the level," video evidence of the dogs attacking your chickens. Then shoot to kill. These cases often seem to turn on whether the dogs were actually attacking livestock and video evidence would erase all doubt.
2) Quietly dispose of the dogs (it sounds like a pack of strays no one would miss) using Conibear traps or even poison. Discretion is often better than being legally in the right since the latter could involve a lot of legal wrangling and bad blood between neighbors.