I personally prefer NOT to use a flotation device, but I am kind of fluffy, and prefer to turn my hands horizontal and use them to generate lift, more than to generate forward motion. There are many different waist or upper body styles.
You should mostly use the regular running motion, it cannot be identical, but the goal is to keep it the same, as best as possible. A little more leaning forward though, as is required to make forward motion.
You should NOT be touching the bottom of the pool. Your goal is to avoid impact (depending on WHY you are water running) although if impact is not an issue, you can if you like.
You do need to make forward progress, be it ever so painfully slow.
My advice is to find a friend to do this with so you can chat. It is mind numbingly boring, and difficult to work as hard as regular running.
You can also try sprints in the water for variety.