I'm on low dose sertraline for decades-long anxiety ]and it has helped tremendously. Prescribed to the right patient for the right reasons, these medication are invaluable. My anxiety has never had anything to do with hormones and never coincided with hormonal issues. I've always known it was biological as I have immediate family members likewise affected with it.
Rebel, I think just taking the medication (don't think on side effects!!) and just going about your day, you will see whether it provides some relief. The relief for me came without 2 weeks of use. And it was dramatic and unmistakable. I know AD's are not for everyone and I don't believe they are the best route if you can definitively say your anxiety is hormone-driven. But for many of us, our history of anxiety dates way back to other issues or causes that have nothing to do with menopause.
This may be why many GPs think about prescribing because, for most of the patients they see (I'm guessing here), the anxiety or depression they experience is not driven by hormones but by other things. That being said, they should be given only if medical issues that could be causing anxiety/depression have been investigated and ruled out.
Patience to answer your question, for me peri and meno have not worsened my already existing anxiety. I don't plan to come off of AD's after menopause and may never stop taking them. That's just me, based on my own history and circumstances. Good luck!