In my prayer time and study I was meditating upon this name. I felt impressed in my own spirit that "I Am" is not only His name but a confirmation of who He is in our lives.
I Am your healer.
I Am your deliverer.
I Am your living water.
I am your burden bearer.
I Am your salvation.
I Am everything you need to sustain you. And so on.
It just struck me that you mentioned this and I wanted to comment. Sorry to be OT.
You're definitely onto something there: God's "I am" pervades our lives ("In Him we live and move and have our being") whether we know it or not. And no need to apologize for "Old Testament" --- all of it is inspired by God, and points towards the incarnation of "I Am," the Second Person of the Blessed Trinity, the Word-made-flesh, Jesus Christ.
He is the Savior of the World and the Messiah of Israel, and He is everywhere, rather prominently in the Old Testament, as all prophecies and archetypes refer in a mystical way to Him. The Psalms refer to Christ page after page and thought after thought, and Isaiah is practically a fifth Gospel.