Mark, the best way to gain a comfort factor with walking bass lines is to think of both hands as one. They need to act together, kick together, and complement each other. If you're soloing, keep the bass simple - maybe arpeggiate the 1-3-5, or even just the 1-5 during intricate parts of your solo. A simple "walk" through the diatonic scale is usually pretty much correct through alot of changes. I played bass more than I played piano, so when I think of the two, I think like a bass player, and just bang out the chords with my right hand. I've said many times that bass is far more important than the chords to me. Tune hierarchy goes like this:
1) melody (usually voice, in my case)
5) everything else (fluff)
Everyone, and everything I listen to is my teacher. Every instrument, every voice, every sound in nature ... they are all my private tutor ... 24/7 for free.