Nearly all running-related injuries are unilateral, affecting one side of the body only. It is true that shin splints are more likely than most other injuries to manifest bilaterally in beginning runners, but what you are experiencing still isn’t at all uncommon. I recommend you minimize your use of the racing flats and, as you suggest, shift some of your running to the bike. Shin splints are typically fairly easy to curb if you don’t let them get out of hand.