There isn't a way to avoid it if you are writing in first person, at least there is none I am fully aware of. However, you can try to keep from starting every sentence with "I" or "my" by changing the wording around a little. Here's an example:
Instead of: "My hands felt warm." You can say: "A warm feeling surrounded my hands."
Another example:
Instead of: "I walked into the room and saw a rose." Try: "Walking into the room, I saw a rose."
I know that probably didn't do much, but it's what I try to do sometimes....