Those continuous-charge fencers are a MENACE, especially the stronger ones. I do believe they may have a legitimate purpose in keeping some *very very escape-minded* dogs confined, but other than that, there is really no reason to put up with the significant extra hazard.
If a pulsing charger isn't working for you, I'm sorry, but it's not the charger's fault, it's because it is the wrong size charger and/or installed/maintained incorrectly. Chook-a-holic's comments are correct. The question is, what voltage was the fence carrying when you tested it with a DIGITAL (i.e. accurate) tester? If it was testing at 3-5,000 v then it WOULD impress all but the most hardened-criminal dog. Most likely however you did not have it set up to carry that much charge, in which case of *course* it did not work, but the solution is simply to correct the installation problems
I am not really addressing this to you particularly so much as to anyone else who is considering installing electric fencing.
The bottom line is, install and maintain it correctly so you have sufficient charge on the whole fence (and train the animals to it so they don't just run through it the first time!), and it works real well. Neglect any of those things, and it sucks
(e.t.a. - I honestly don't know whether continuous chargers are supposed to be run at lower voltages, but I can tell you that 600-1,000v on a proper pulsed charger will not deter ANYTHING that actually wants to go thru the fence. You need like 2,000-3,000 to reliably contain cattle or horses, more for dogs and predator exclusion. USE AN ACCURATE I.E. DIGITAL FENCE TESTER -- those cheap 'five neon lights' jobbies are terribly terribly inaccurate)
To the o.p., on the question of killing squirrels and small birds, even pulsing chargers CAN do that if the animal is sufficiently small (esp if it is wet at the time) and it contacts the live wire AND a metal fencepost at the same time. This is uncommon but does happen, if you use metal T-posts. I would not particularly worry about it with chickens, though, as they do not like to try to fly up to perch on something as thin and weebly as a wire that they can't see well. It is not *impossible* for electric fences to kill chickens (main cause: chicken gets stuck in fence and thus zapped repeatedly, or person was using one of those continuous-current chargers) but it is quite rare.
Good luck, have fun,
Pat