I honestly think that would come down to how you best learn things. Like if you learn anything by breaking in down in to small sections and doing one step at a time then I think you would benefit from doing one cross/side at a time. However if you learn things better my repeating the whole thing/steps over and can do that (like I do) then I would say to start on whatever side you seem fit.
And although at first it may be a little annoying to be getting slower times and you may want to start on a color you are familiar with, I would say to try and stay focused on becoming full CN and don't let "slower times" at first discourage you from doing what it takes to become full CN.
Also, depending on when you next have a competition, I would try to be fully CN before your next comp, especially since you can easily be full CN in less then 1 month (or even less then 2 weeks with a bit of practice).