If the airlock started bubbling within an hour of tossing in the package of yeast, it is not the new yeast that are doing it - instead, you are seeing gas release from the other yeast which are now working. In any case, I'm glad to here your batch is off and running.
And just for info, you can stir stuff in a carboy, and in fact during the first couple of days you probably should to aerate the yeast. I use the handle of a plastic stirring paddle - it fits through the neck of the carboy and allow agitation. I have also used a short piece of thick walled tubing that doubles as a blow-off tube. Some folks have the fancy lees stirrers that attach to a drill, but I haven't felt the need to spring for one of them up until now.
"Our results are merely the result of carefully managing the transformation of bee spit into yeast excrement."