A few questions: Is he active (walking, running, working out, biking)? How is his diet (good real foods like fruits, vegetables, lean proteins, or fast food, food from a box, beer and potato chips kind of snacks, lots of ice cream, etc)? Is he employed and satisfied with his job or unemployed or unhappy at work? Is he overweight, or does he have limitations due to back pain or headaches? Has he had a checkup for heart problems, diabetes, general health? Does he have stress, like finances, an ex who causes problems, the roof leaks, his car got repossessed, whatever?
I worry about a couple of phrases in your post. The first one is that you're "not really pill people" (whatever that means!). Usually, people would rather not take pills. We'd all like to be healthy and to have our loved ones be healthy. But disease happens, even to the seemingly healthiest people (that whole "ice bucket challenge" that was all the rage this summer was started to honor a very young super-star athlete who did all the "right" things and was in better shape than most of us will ever be and who now has that horrible disease and is nearly completely paralyzed). Medication can help. That doesn't mean that you pop every pill known to mankind, but you seek competent help, take the best medication for you in moderation, make sure your lifestyle is as healthy as possible, comply with your doctor, and appreciate modern medicine and research and how it can help you. Yes, you and your SO should worry about his taking someone else's prescription. That's the worst kind of "pill people". You can't call yourself "not-pill-people" and at the same time take pills intended for someone else. So you are "pill people".
Secondly, doctors don't "hand out" pills. Responsible doctors write thoughtful prescriptions when they believe that medication can provide relief to their patients and that their patients will take the pills as directed and will do their part to become healthy (stop or decrease drinking, stop smoking, lose weight if necessary, eat fresh fruits and vegetables, get enough rest, etc). Perhaps your doctor isn't "keen on handing out pills" because your SO borrows other people's prescriptions and therefore hasn't demonstrated a proper medication ethic. My daughter takes many medications and she is required by her pain specialist to sign a form that promises that she will not seek multiple pain doctors for multiple and duplicated prescriptions, that she'll do her part to take responsibility for her health, that she won't skip or double up on dosages, that she won't share her medications with anyone else.
Another point: some (not all) anti-depressants don't treat anxiety. If your SO is having debilitating panic attacks, he needs to see a psychiatrist who can perhaps provide him with a prescription but also work on lifestyle changes, relaxation techniques, counseling, etc. A psychiatrist can determine what will help the most, and it might be a medication. That's nothing to be ashamed of.
Please encourage your SO to stop taking other people's pills and to seek professional help. A competent doctor will not just drug him up, he will use a variety of tools, which may include medication and lifestyle changes and counseling, to make sure that your SO can function in a healthy way. It's important to take any new medication for the required amount of time to allow it to take effect, and to take it properly (at a specific time, with or without food, etc), and in conjunction with whatever else is needed to become well.