Had this problem with my 704 wifi a couple years ago when the original battery went kaputz. Replaced the battery and the problem went away. Seems if it really can't hold charge it can't muster enough juice to initialize the screen properly. I had some of the anomolous behaviour mentioned in the thread while attatched to the docking station also so having ac power won't necessarily help if the battery is really burnt or the cells are damaged.
New batterys are available on ebay and at newmp3technology.com. You shouldn't need to spend more than about $30 US.
I agree with above comment. I find mine white screens sometimes on startup when the battery starts to get low. I have a 705 as well and have no problems with that so when this happens I swap the batteries and then charge the offending battery on the 705 and it seem to solve the problem for a while.