Seems you already know what the problem is, so I'd take a just-do-it approach. Just start playing chords with your left hand instead of breaking it up. This is a whole lot easier than it seems.
On piano, I chord with both hands, moving my left hand around the bass. For you, since you seem to be on the beginner level, theory-wise, just use the chords you know. They can do more than you think they can. Also, try playing lighter chords with your right hand, until you get the left hand down, using just two notes or the melody. This will give you a jazzier sound, especially if the left hand is playing two or more notes with the right hand.
Buy some books on piano jazz chords. If you don't know the scales and progressions numerically, learn it. This will help you understand those books. Everything else is built on what you know already, so look at your situtation this what: "I'm poised to become a great musician, all I need is practice". Do those things. Analyze your hands when you play something that sounds new to you. You'll start to see patterns. Practice religiously, at LEAST 2.5 hours a day, everyday, and you should see this goal sooner than you think.
I don't practice everyday now, but I usually get in 16-24 hours by the end of the week. Stick around, let me know your progress. Try asking more "why" or "how" questions, instead of the "can I have" questions that are usually posted here.
