I don't know what jurisdiction you're in, but ordinarily you can seek punitive damages in a trespass action in addition to the actual damages causes by the dogs (i.e. dead chickens). I'm not a lawyer and not giving you legal advice, but generally speaking trespass is a tort where damages is not an element; meaning you don't need to prove that your property was damaged to win. Generally speaking if you win, you get costs, but be careful, expenses like developing pictures, purchasing police reports, etc. may not be costs taxable to the losing party in your state.
Something to think of might also be to seek an injunction along which your request for money damages, i.e. a court order to keep those dogs off your land. Any subsequent dog problem would then be contempt of court... police could enforce the order, your attorney could get his fees from the dog owner, etc.
Generally most local bar associations have a lawyer referral service and allow for a free consult... it's always worth it to hire a lawyer before resorting to shooting at anything!
There are a number of organizations that provide free or low cost legal service & your county bar association might be able to put you in touch with them.
I understand wanting to shoot a dog, i have been tempted myself... but its not the dog's fault, its the owner who should be brought to task. (And you don't want to risk breaking the law over a chicken... even if its a nice chicken and not a mean rooster).
