Had the same problem here ( just putting one back together now). Likely a PCH issue, so basically a case of dumping the board and cutting your losses. This model is having a lot of this issue. Warming the PCH up at 150~180 probably might give you another couple of days but it'll fail again soon enough. It's NOT likely a ball problem, rather the issue is in the internal gold connects of the chip itself.
I suspect it'll be because the little black rubber shroud that sits over the proximity sensor has fallen off when you did the cable transfer. I had a client phone come in the other day and I noticed it was missing ( from a previous repair ) and asked if they'd been having issues, and they have, with poor face-presence recognition and similar. To date I've not yet found a supplier of replacement units.
Same has happened here, nicely smashed front glass, replaced the whole frontal assembly just to be "quick and easy", alas to no avail, touch screen doesn't respond, tried with old/original unit, still doesn't work either. I fix a few iPhones/iPods a week and 3G (and 3GS) units are usually the simplest of them all, but this time it looks like it might well be an onboard controller fault :(
The problem is quite common. It's the fault of one of the main NEC/TOKIN capacitors located directly beneath the CPU (on the other side of the board). The reason why it works on battery but not AC is because there's no significant ripple in the power when running from battery, but on AC there is a fair amount which causes unreliable behaviour. Replacing the capacitor is not a trivial task even for those who deal with surface mount replacements. Typically we just break off the old TOKIN capacitor and install a set of tantalum caps in its place.
Try this solution... Press and hold the home button, then the up-volume button, then the sleep/power button (so all three buttons are now pressed and held) --- it takes a while, but the white screen will go away and then after another 10 seconds or so (still holding btw!) the apple logo will come up, release the buttons and then wait, it'll boot normally again in a minute or two. EDIT: Just tried it on my own 3GS here which decided to go into the WSoD - alas, doesn't seem to be working this time, not sure though if it's the buttons or not at fault. Update: If the home-up-sleep combo doesn't work ( as I encountered recently ), it might be worth letting the unit run flat for a few days and then plug it in and try the home-up-sleep combo again. I was surprised, I thought I was going to have to bin this 3GS but amazingly it came back to life for me. Not sure what the difference is, but perhaps during the total power-down phase the iPhone manages to go through some different stages in the firmware, as opposed...