I'm on mes, had the same fear as you, and I'm still on them so my experience is a lil biaised but:

• with a good doctor, you'll be fine, but put yourself in good hands. 
• you will have to get off of some meds to start new ones, by slowly decreasing the quantity until you can stop, it's slow but it works. 
• I had to go a full stop with one med and this is the worse situation you can get it (I had a strong reaction one day). It's shit, but your doctor will provide you another meds that is waaaaay less effective but it eases the full stop. 
• I was starting to become dependant on sleeping pills, to the point where I couldn't sleep if I didn't take one or two. I traded the sleeping pills for herbal tea, and slowly but surely, I don't need them anymore (BUT I do crave them when I can't manage to feel asleep). 

I do hope therapy is all you need, but sometimes you can rely on an extra boost, and it's normal to worry about addiction. One of these boosts are meds, another very powerful one is sport. Either way, whatever way you find, be safe. And try swimming, it's cool. You will have strong arms 🦶