Hi. I've just finished first year medical school so I'm not hugely knowledgable about what a career in medicine actually involves but I have a few points that may be of help.
First of all, we are constantly being told at my med school that medicine needs a whole variety of people of different personalities with different strengths and weaknesses, and different abilities to work under stress/lack of stress. Some medical students will go on to become surgeons who perform operations which are rarely successful, or will go into palliative care where many of their patients will die. But not all parts of medicine are like this. There's a whole wide range of specialties which avoid a lot of death. Plastic surgery can change lives (e.g. of burns victims, people with really low self-esteem due to issues with their appearance) but many of the patients will not have terminal conditions. Likewise there are specialities like ophthalmology and others (sorry can't think of the top of my head) in which many of the people will be relatively healthy. Of course you'll undoubtably experience much death and pain and suffering along the way (e.g. as a junior doctor etc) but the end result of you degree doesn't necessarily have to be dealing with people who are in immense pain/are dying if you think that that is something you personally would find very upsetting/difficult to cope with.
Secondly, as sad as it seems, you do become desensitised to these things as you go along. At first I was mortified by the idea of cutting up a dead body, and I couldn't stop thinking of how they had once been a person with thoughts and feelings, but now I don't think about it so much, and it's just another way of learning. I also will often be struck when taking a history from a patient of how hard their life is/how hopeless their situation is and think that i'll never forget them (i also felt like this when doing work experience in a hospice prior to applying) but I imagine that after a while, it becomes impossible to remember every patient you have contact with, and you end up just making as much of a positive different you can to each individual before moving on to the next. This may not be the case (doctors/more experienced medics, feel free to correct me!!). You're also taught in medical school how to cope with emotions, and how to reflect on incidents and make yourself cope better in the future.
Finally, no one can decide if medicine is for you, except you. I would recommend getting some work experience in a GP surgery and hospital as well as a care home to give you a wider idea of what medicine as a career involves. But when push comes to shove, I would say go for it. Being sensitive isn't a bad thing - it will help you be naturally empathetic and good at communicating with your patients and making them feel at ease. But if you find it really difficult to cope with, don't put yourself under unnecessary pressure and emotional distress - there's lots of other worthwhile degrees and jobs out there that you might be better suited to!
Sorry for any typos, and if you have any questions get in touch (oh and also I have a blog you may be interested in reading - link is in my "About" section