Of course you can get better without medication. Medication is serious business, and everyone should avoid it if possible -- artificially altering any part of your natural body processes should only be done if necessary. So by all means try to fix it through therapy. Meditation should be done regularly, not just when you're feeling calm -- the whole point is to watch your thoughts so you realize they aren't you, they are just part of you. But the above is correct, it helps to see a therapist. If you want to try natural medicine then an herbalist or naturopath or practitioner of integrated medicine might be helpful, but if you want to solve this through changing the way you think then try CBT, a form of therapy. Good luck. And if it doesn't work and your life is seriously disabled, then medication will still be there.
It IS possible to overcome anxiety without medication, but I would strongly recommend you work with a good therapist if this is the path you choose.
It is absolutely your right to decide you do not want to take psychotropic (NOT "psycho") medications, but where did you get the idea they would "screw your head up?" For generalized anxiety, which in my humble and non-medical opinion is what I believe you have, the medications used are very effective, mild and safe. With most of these meds, the worst side effect is being tired until tired until your body adjusts to them. I think you may have Googled medications for much more serious and debilitating forms of mental illness than what you are experiencing.
Using the correct medication for the shortest time possible can be an amazing tool to help you deal more comfortably with your anxiety as you move through your therapy.
You can visit our Alternative Medicine Forum and get some good advice from the folks there. A Naturopathic doctor could also be helpful for you.
Whatever you decide, I wish you the best and relief from your anxiety.
RubyWitch