It sounds like you are going into this having already diagnosed and prescribed yourself. Although I don’t doubt you may be depressed and have anxiety, it’s generally prudent to consider all available treatment options that a medical professional recommends. Cognitive behavioral therapy attempts to address the causes of depression and anxiety, which matches your desire to deal with the issue, not just cope. My main point being, see if the doctor recommends meds before you explicitly ask for them.
Hope this didn’t come across as too preachy. I’m sorry you’ve experience anxiety and depression for so long and am glad you’re seeking medical assistance.