I wouldn't stir before racking into the bottling bucket as that will get a lot of yeast and trub back into suspension.
Stirring, in a smooth, gentle, motion so as not to aerate the beer, after racking onto your priming sugar solution should ensure that the priming solution is evenly distributed throughout the batch.
I take it you don't have an autosiphon. If you have a bucket of sanitising solution sitting around just put the hose in there and let it fill up, say slightly above half way, then put the end with most air into the fermenter, keeping the other end closed off with a sanitized finger and below the level of the fermenter. When you're ready to start racking, put the top end of the hose into your fermenter, release the bottom end over the container of sanitizing solution and the solution will start to flow out, pulling beer into the upper end of the tube. After all the sanitizing solution has flowed out, close off the end of the hose with your finger again, or very quickly place the end of the hose in the bottom of the bottling bucket so the beer swirls around rather than splashes from a height.
BTW, here's a video that might make up for my crappy explanation
I know I've seen a better one somewhere but couldn't find it. I'm sure you'll get the right idea.