I don't have a wall charger, and my phone charges significantly faster while turned off. However, if I shut it down while connected to my computer (or connect it while already shut down), it will immediately turn on.
In this thread, the user was instructed to change USB Computer Connection behavior, but the instructions don't make sense on my phone (probably due to an OS update, I guess). In particular, he was told, "You have options of Mass Storage, Media Device, or Camera," while currently only the latter two options exist. Worse still, the explanation was that switching away from mass storage (the option that no longer exists) would fix the problem.
Please help.