Fog and haze
Thinking about all the times I've seen fog on the summit of Mount Washington, I started wondering why air that is very hazy at the bottom does not always become fog at the top. Haze implies a high moisture content, so I would think that it would cool to its dew point as it rose, resulting in a cloud, if you could see it, or fog, if you were in it. My knowledge of meteorology is rudimentary, however, so I won't speculate. I would guess it has something to do with acrobatic lapse rates and erogenous cooling.
I'm not sure how acrobatic the lapse rates are, but they do tend to be adiabatic. And yes, the simplest answer to your question has to do with lapse rates.
Everyone knows that hot air rises, cools to its dewpoint and forms clouds. Air continues to rise as long as it is warmer than it's surroundings. The reason your hazy parcels of air are not condensing into clouds is because they are rising, and cooling faster than the surroundind air. So they just sink back to where they started, never getting cold enough to condense the water vapor into clouds.
The difference between fog and haze, is that haze forms above the dewpoint temperature, because a chemical complex is formed between water and various hygroscopic substances. These substances can be anything from sea salt to pollen to topsoil to manmade pollutants (e.g., acid gases). I once had the privilege of climbing out of an 1800-foot-thick haze layer, passing through about 50 - 100 feet of brownish-yellow vapor (which I'm guessing was nitrogen tetroxide from a local coal fired power plant...the brown vapor plume extended downwind from the power plant a good 40 miles and vanished upwind of its stacks), ascending through another 2000 feet of clear air with upward of 50 miles visibility, then entering a dense layer of altostratus and emerging on top in clear air and sunshine. So, yes, haze is denser than water clouds because heavier substances are present in it, which causes haze to hang near the ground and keeps the air at higher elevations relatively clean. (Noteworthy exception: The 1980 eruption of Mount St. Helens sent a dust cloud 30,000 feet aloft...in which a haze formed wherever sufficient humidity was present. But that just tends to reinforce the point.)