My daughter developed the same problem on her 2006 MacBook Pro. After trying all kinds of suggestions gleaned from online, we went to the Genius Bar. Although the laptop passed the hardware diagnostic test, it was apparent when booting the laptop from an external drive that the problem was still happening and thus was a hardware problem with the trackpad. More specifically, the trackpad thinks that its button is depressed at all times, leading to the 'grab everything' problem.
We could have gotten a new top case (can't get just the trackpad) for $250 plus labor, but since my daughter is 12 and mostly uses the computer to synch her iPod and play The Sims 3 (and the computer is 5 years old), we decided to go with the $10 workaround of buying an external, wireless USB mouse and disabling all input from the trackpad when a mouse is plugged in, under Universal Access (in System Settings). She is happily playing the new Sims Generations beside me as I write, and her good old MacBook Pro lives on to fight another day!
I have to say, I spent most of a day trying to determine whether the problem was in the hardware or the software (we had even tried an external mouse, but hadn't known to turn off the trackpad so it seemed as if the external mouse also had the grabbing problem, when it was just the trackpad button always thinking it was pressed - very confusing). At the Genius bar, we were able to find out where the problem lay in about 15 minutes, so I was glad we asked. Thanks to Jason at the Nashua, NH Apple Store.