A lot of heals can't be used while running, it's not just Shaman heals. If you need on the move healing, you have Riptide, Ancestral Swiftness, Spiritwalker's Grace, Healing Tide/Stream Totem, Unleash Elements and Earth Shield. Every class also has at least one means of self-perservation that you should also be making use of to reduce the need for healing.
It sounds like you're expecting multiboxing to be a lot of facerolling - there's a lot of
work,
setup,
practice and gear involved in making it look easy. "Look" being the keyword. There's a lot more
hair-pulling,
keyboard-throwing,
gremlin-squashing and carpel tunnel involved than most YouTube videos tend to show.