Rephrase things so that you don't need to start sentences with "I":
"Is this a death sentence?" was the first thought that occurred to me. Death scared me to death. The exam table was directly facing the doorway, and I was sitting at its end. I wanted to run away, far away. As I jumped off the exam table, my legs gave way, and I fell back onto it.
What I did was use other things as the subject of the sentence (and "I ..." became "... me"), or introduce sentences using prepositions. That isn't to say that as few sentences as possible should start without "I" (I couldn't find a way to change "I wanted to run..."), because that would be just as stilted.
If you're unsure what you've written flows well, read it aloud. I think it's more likely that your eyes will skim over repetitions, but your ears will catch them.