I thought I should come back to this as I seem to have caused a lot of the controversy on this thread.
The rules are clear. You don't have to put your dog on a lead on country paths unless 1. wild bird breeding season 2. you are in the vicinity of livestock 3. your dog cannot be kept under close control off the lead.
Clearly the dogs on this path are all in the vicinity of livestock and some of them cannot be kept under close control off the lead.
So yes, the dogs should be on the lead. Not everywhere in rural areas, but on the section of path by the OP's property they should be. Clearly dog owners should abide by the rules.
In reality, this will not happen. The Op can't negotiate this as it involves lots of people, many of whom she cannot identify. So the OP needs to do something.
Shooting is not really an option because there are lots of rules around it, involving whether or not the dog owner is/is not present, can/cannot be identified, can/cannot get the dog back under control when challenged by the farmer, the regulations and issue purposes of the individual firearms certificate. Even if the OP is vindicated in a court case when she shoots someone's dog, I doubt she really wants to go to court or shoot dogs.
So the common sense solution is to either move the chickens, improve the fence, or change the wording of the sign to something people will take notice of. I would personally go for a sign saying something like: 'Number of chickens killed or maimed by dogs off leads past this point: 17 (or however many). You must put your dog on the lead in the vicinity of livestock.' I would also go out and talk to dog walkers, particularly those who have got dogs on leads, and ask them to speak to other dog walkers they meet on the walk about the problem. The Op may already have done this, and have harsher signs; I don't know.
I think working through the well behaved owners is a better way of approaching it, because it will make the non-lead users feel policed on the walk, which the OP cannot do alone. The public are not out to get farmers; myself and my neighbours have put livestock back in fields, phone if we see a lamb struggling, speak to people misusing rights of way etc.
It is similar to car drivers parking illegally. I put DD in a reflective safety vest. Morally and legally I don't have to. Morally and legally cars shouldn't turn around on my land, or park in no parking zones next to DD's school, but they do.
What the OP can practically do is more important than legal and moral positions that cannot be enforced in her predicament.