I always use a socket, out of fear of overheating the chip upon soldering. I know full well that I can safely solder just about anything without frying it, but still do it anyway. Sockets are cheap, and you will be happy with yourself for using one.
I too have never had great success with desoldering braid. With the cheap (manual) solder sucker that Radio Schack sells, I can safely and quickly desolder any DIP chip without damaging it. Buy two of them though, they break easily.
The trick is in the technique:
1) Re-tin each pin before attempting to desolder.
2) Heat a pin just enough to completely melt the solder on it. This takes practice to get just right.
3) When the solder is melted, remove the iron, and quickly put the sucker all the way over the pin, squarely against the board, with pressure; and push the release button. If you keep the sucker squarely pressed against the board, and the tip of the sucker is in good shape, most, if not all, the solder will be removed.
4) Use a solder tool, or a screwdriver, to make sure the pin is mechanically free. If it's not, keep pressure on it, and apply heat with the soldering iron. When the pin comes free, wiggle it as the solder hardens so that it doesnt get stuck again.
This isnt as hard as it sounds, and can be done very well with patience. I have no trouble desoldering and re-using the most sensitive chips this way.
It will help if you cut the chip off the board first, but I usually don't bother. It's easy to damage a board doing this anyway.
Also, desoldering on a PCB should almost never be done with a heat gun or 250W soldering gun. I couldn't tell you how many times I've had to repair a board when someone else did that.
