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Hi there! I have read your question and hope to be of some assistance to you today!
I find it interesting that we are just about the same age, and that I, too, am entering menopause. The physical effects of hormones on our emotions is real, and the physical effects themselves can scare us when we don't know what is happening.
The things that you describe, palpitations, etc. are quite common. When you don't understand why it is happening, you experience the feeling as a fear of the situation you are in when it happens (such as being at the store) and you then connect the feeling to being in a store, when in fact, it is just due to a rush of hormone that just happened to occur while you were in a store! It is the fault of the hormone and not the store--- but our minds connect the two, and your job now is to unconnect the feeling of the hormone rush from normal daily life.
Side-effects of medication are a real thing, as well. There are some wonderful resources out there on how to manage physical anxiety without medications, such as this one: http://www.helpguide.org/mental/generalized_anxiety_disorder.htm
Getting daily exercise, changing your thought patterns and self-talk from being fearful to remaining calm and breathing deeply can really help, as well. Knowing that this feeling is simply similar to the experience of PMS (when sometimes we feel cranky and irritable, but blame it on life circumstances instead of just accepting that it is our body going haywire for a bit :-) ) can also help... and knowing that you are totally not alone in this experience, and talking to other women who have experienced it can help, too!
However, if you choose to try the medication, just tell yourself you'll try it temporarily. Try it for a month or so and see how it makes you feel, and talk to the Dr. about that if it becomes uncomfortable. And research menopause/perimenopause, as well to get information on this stage of female development. You may just relax after reading about it and knowing that you're in good company with thousands of other women who are going through it with you! It is your body, your life and your choice to handle this period of time however you feel most comfortable with. I hope that the link I shared will give you some ideas of how to calm the mind, and let me know if I can be of further assistance to you here! I wish you all the very best!