Hi - after reading your comments, it might be better to start another comment thread. I remember hearing things that others could play (albeit Coltrane and such), and really loving the tension and sound, knowing there were underlying mechanics but not knowing anyone who could fully explain it in a way that I could do it myself.
Not to be presumptuous, but given what you've left out, in sounds like you may not have the fundamentals of diatonic harmony sussed to the point where you can both play and hear it in your head. If this is true, one approach you could take is to start by analyzing existing music, like, "All the Things You Are", for example. There is a structure to this all, and it's not about the left hand vs right hand, or even being able to play one specific chord.
Songs have an arc, like a story. This arc gives the listener a sense of movement and stasis, like that feeling of completion when you played the "...fleece was white as snow" portion of Mary had a little lamb. I'm a bit hungry right now so I will use food metaphors, after all, what is closer to the root of your being than food and music, amiright? Learning a particular chord is only like having a spice in your cabinet (most chords are just variations or intensities of their basic form). Knowing recipes, how much and when to apply spices and what spices to substitute for others, or variations - sounds like what you're after. I think learning from some basic recipes is a good start, but taking the time to analyze what you are playing. You have to get concepts under your fingers. "Knowing" a recipe is very different from making a great dish.
So, music recipes. In your recording, your flavors were of I, IV, and V. If that is foreign to you, it denotes the harmonies related to degrees of the major scale (sorry if I'm being obtuse. The beer is strong). That means in the key of C MA, I = C Ma, IV = F MA, and V = G MA or Dom7. When you move from the IV to the I, or the V to the I, you have a sense of "going home" or "completion". Let's see if we can make the journey away from and back to home just a tad more exciting.
We are more than a bit limited by this interface, but let's try.
In the key of C MA, let's try that lamb song, but with a different approach. Don't play chords. The right hand plays the melody, the left hand will play single notes, all quarter notes on the beat.
Left hand will play these notes: c, d, e, f, g, g#, a, g#, g, f, e, a, d, g, c. The right hand plays the normal melody (primarily 8th notes).
Let me know if this works for you and we can continue by building on this.