There is a particular style of art that I would like to use in my games, but I can't manage to pull off.
Here's a really good sample of what I want to be able to do:

https://orig00.deviantart.net/4955/f/2010/098/8/c/texture_by_torbak.jpg

It's a hand-painted style, somewhat cartoonish but still containing detail.
But my attempts to paint such a texture result in a lot of solid-color blobs.  I can't ever seem to convey shape without drawing an outline.
I know I can't expect this to be "easy" but I can't ever quite get a grasp on what I need to do to my drawings to get them to look right.
I feel as if I am missing out on some sort of technique or tool, or maybe some sort of fundamental understanding.  I would greatly appreciate it if anyone knew of a good tutorial I could follow so that I could practice these skills.

You might check out the polycount wiki for hand-painting stylistic textures. Googling for hand-painted and stylistic texture tutorials might net you a few additional forum threads and/or articles. Try polycount, zbrushcentral, blenderartists, etc...

Edit: Might check this mini-tute thread by an artist working on one of the World of Warcraft expansions. Has a whole bunch of neat images to look at, plus some tutorial discussion later. WoW artists are always fantastic resources for learning about stylistic textures.