Your system is low on R134A, either from a leak or age, the stumble every few seconds is the compressor turning on and off, there is a low pressure switch on the system that makes it do this to keep the compressor from damaging itself.
I just paid 45 bucks and they vacuum tested my system, told me I had no leaks, then they evacuate the entire system of old R134A and add new stuff. They can't just add to the old because they won't get the amount correct, there is no way to tell if the right amount is in there unless they take all the old stuff out. I think the machine that did my car actually recycled the old R134A. Either way, for $45 bucks, my AC is ice cold, and the compressor stays on for a much longer time, so the stumble feeling is gone, although it still will kick on and off once in a while, but it won't be ever few seconds like it is now.