In general, anxiety can be handled with medications or with "talking therapy", of which there are many types. So far you have seen two doctors.
One offered medication. You were concerned about depending on them rather than learning to handle situations yourself, and about possible side effects or damage to yourself from those pills. These are wonderful things to discuss with that doctor. You can ask about side effects, you can ask if the doctor thinks you need to be on the medications indefinitely or just while you learn to handle difficult situations, and so on. You can also ask about needing higher doses for more difficult situations. Once you thoroughly understand what the doctor is suggesting and why, and the benefits and risks of it, you can decide whether you want to take it and how you will know whether you want to keep taking it or not.
One recommended just getting healthier overall. A great question to ask that doctor is why that might help? Perhaps you're sleeping very poorly. Perhaps some blood tests have shown you are suffering from deficiencies that might cause anxiety. You need a reason to believe that exercise and healthy food will help the anxiety, right? That said, living a healthy live often means living a happy life with less worry, so if you can afford (time and money) to walk more, spend time outside, eat fresh food you cooked yourself, and take up an active hobby, I would encourage you to do that as a good thing in itself that might happen to lower your panic situation too.
In addition to asking more questions from the two doctors you've already spoken to, you can look into counseling or therapy. There are many kinds! CBT is often suggested for anxiety - rather than discussing your childhood and trying to find out why you react as you do, it focuses on teaching you specific skills for coping with things that distress you today. Many people learn what they need to learn from such counselling and then stop doing it. The Wikipedia article includes links to studies, including one specifically about adults with anxiety.