One of the main factors for growing crystals is there needs to be what is called a nucleation site. Some rough spot for the crystals to form on and build off of. A "rougher" string will have more of these nucleation sites, a "smoother" string will have less nucleation sites, and thus the crystals should grow ontop of already formed crystals more then just forming all over on the string.
A smoother "String" is fishing line, but I do not think it will work in this case. Go to a Walmart or some store with a sewing section (take a magnifying glass) and just look at the strings and see if you can find one that appears "smoother" and try it. This might not effect it that much though, but worth testing out.
Dilute solutions will form crystals slower. Slow formation of crystals tends to make larger more pure crystals with less imperfections and could reduce the "cloud" like structure you are talking about. But you have to be careful about letting the solution dry.
You can maintain a constant liquid level by using an inverted coke bottle. Like you often see it dog water bowls and such:
http://www.dinodirect.com/water-food-feeder-dog-cat-pet.html?cur=USD&AFFID=183