I just found out about a better methodology after already writing thesis - how should I prepare for my defense in 3 weeks?
This situation is one of many that demonstrate the key role of an academic advisor and why I would never suggest anyone attempt a research program as a novice without one.

The specific content of your research (i.e., the merits of content analysis vs. critical discourse analysis for your research question) is off-topic for Academia.SE, but at this point you should find an advisor in your field to ask for advice.

If you cannot reach your thesis advisor, then maybe someone else on your committee can step into that role. Your advisor has approved your work, so either they have not read it carefully (and have failed you miserably in their advising duties) or this problem is not as serious as you may think (i.e., your choice can be justified despite your oversight of another technique in the field).