Let's take them one by one.
1 - "moving" vs. "leaving": Both words are fine, they just have different meanings. Neither is better or worse. It just depends which idea you want to express.
2 - "I've been staying" vs. "I moved back": Again, both are fine, they just have different meanings. Neither is more natural. You just have to decide what you want to talk about. Which is more important to you, the fact that you moved, or the fact that you are staying there?
3 - The second version is more emotive and personal. It conveys more feeling. That is because of expressions like "in my life", "even my life", "inspired". "My health" is more personal than "one's health". However, "cherish" is a beautiful word and "appreciate" is better than "appreciation" because verbs carry more force than nouns.
4 - I don't like either version due to their excessive use of present participles. "I've decided" is better than "I decided". "Dream" would be better than "dreams". "To study" would be better than "of studying", and "travel" would be better than "traveling". As for the last sentence, the word "prepare" doesn't make much sense to me. Prepare what? You can say it like this:
"For this dream to come true, I must take steps to allow it to be so. I'll need to care for myself that I might continue to recover, and I'll need to give myself time."
5 - Neither is better. "Moving back" is better than "getting ready" because "get" is almost always a lousy word. Also, "moving" creates an image while "getting ready" creates none. "Starting to work" and "start working" both waste energy with present participles that are not needed. "Dreams" is better than "goals" because you have been talking about dreams, not goals. I would say it like this:
"Soon I'll return to my own place, there to start, one step at a time, towards my dream."