I don't think any doctor should tell you you should be on meds purely because it's in the family. That doesn't mean in any way you will need meds for your whole life. I have anxieties similar to yours. Death scares the hell out of me and I'm constantly finding new symptoms to google. I've been having anxiety attacks about death and the eventual nothingness since well before your age and I'm only 23 now.
Maybe you should see a therapist and try and work out your anxieties? From what you wrote here, I don't think that you need to be on meds the rest of your life.