MacBook 13" will only charge battery when on and open. why?

Question

Hello,
My white 13" Macbook, model number A1181, will only charge the battery when the computer is on and open. If i close the screen down, the light on the AC adapter will go off about 2 seconds later, and will stop charging the battery or giving any power to the computer. When i try to start up the computer without the battery in, it will not start up.
When I attach the AC adapter to the computer and it is not open, or not on, there is just a faint green light on the AC adapters LED light. About 1/4 the normal brightness.
Also, a few days ago, the computer would randomly shut down when it was plugged into the AC adapter to charge my battery. I was surfing the web and it shut down three separate times. All while the AC adapter was in. After i took the AC adapter out, it stopped randomly shutting down.
I went to a friends house who has the same mac laptop, and have swapped out his battery and his AC adapter with mine in every combination and everything operates in the same way: not able to turn my computer on with his adapter plugged in and the battery out and a 1/4 as bright green light when the AC adapter is plugged in and the computer is off or the screen is shut. My AC adapter functions perfectly with his computer, as does my battery, perfectly normal.
I thought it might have been the DC in board, so i replaced that myself with a part i ordered online, and it was not the DC-in board, same problems persist.
It seems it must be something on the logic board that has shorted out or broken that routes the power when the computer is without battery, shut, or shut down. and there must be a separate line going to the battery charger, which is why that aspect of the computer is still able to function. this means however that my only option to keep using the computer is to keep the battery relatively full and NEVER let it drain anywhere near low, as i would not be able to boot it back up again if it fully drained.
Is this problem indicative that the whole system is on the way out? or might i get a few more months out of this thing by just charging the battery while it is open, and while im working? Any idea what this problem might be? It is past even the 3 year extended apple care plan we had, so that is not an option.
Saw someone post something similar to this in another thread where the power information was requested. Here is mine:
thanks for any help
Battery Information:
Model Information:
Serial Number: DP-ASMB016-371d-4927
Manufacturer: DP
Device name: ASMB016
Pack Lot Code: 0002
PCB Lot Code: 0000
Firmware Version: 102a
Hardware Revision: 0500
Cell Revision: 0102
Charge Information:
Charge remaining (mAh): 4468
Fully charged: No
Charging: Yes
Full charge capacity (mAh): 4771
Health Information:
Cycle count: 307
Condition: Good
Battery Installed: Yes
Amperage (mA): -737
Voltage (mV): 12333
System Power Settings:
AC Power:
System Sleep Timer (Minutes): 0
Disk Sleep Timer (Minutes): 0
Display Sleep Timer (Minutes): 25
Automatic Restart On Power Loss: No
Wake On AC Change: No
Wake On Clamshell Open: Yes
Wake On LAN: Yes
Display Sleep Uses Dim: Yes
Battery Power:
System Sleep Timer (Minutes): 0
Disk Sleep Timer (Minutes): 0
Display Sleep Timer (Minutes): 15
Wake On AC Change: No
Wake On Clamshell Open: Yes
Display Sleep Uses Dim: Yes
Reduce Brightness: Yes
Hardware Configuration:
UPS Installed: No
AC Charger Information:
Connected: Yes
Charging: Yes
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Answer 1

You seem to have narrowed it down to a logic board issue. You can try resetting the SMC:
http://support.apple.com/kb/HT3964?viewlocale=en_US
And the PRAM:
http://support.apple.com/kb/ht1379
But it sounds like either an expensive repair or a new Mac is in your future. Make sure you're keeping regular backups.