To my mind the framework is linked with the ethos or ideology of the researcher in question. If you think critical discourse analysis method is the best then go ahead...Of course I do not need to remind you that such a theory is generally leftist and struggles to find injustice and inequality in roughly every analysis of a social phenomenon. You have Michael Foucault and his famous power theory. Or Derrida, Barthes-you name...Truth be told, an objective analysis has to be mixture of all these schools and more. Good luck!
Excuse me - Does CA means California or Canada? I think your method is sound. Identity construction is not a topic that I am well-versed in, but an ethnographic study on that I might be able to speak into. Does CA mean California or Canada; then, I can focus on your question at hand...
Have you looked at Hall & Buchholz's Gender Articulated? Great chapters in there by Susan Gal and by Herring, Johnson & diBenedetto. Lots of constructs you could use for analysis. Also, Digital Discourse by Thurlow & Mroczek.