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#11
The two crash reports are bugcheck 124, simply more of the same. A 124 error can be caused by drivers. I know you updated drivers. Did you update those Conexant modem drivers? They did not show in your latest dumps; but then the crashes happened so quickly, they might have had a chance to completely load. Do a search of your C: drive for each of those drivers in red font in my post above. When you find them, look at the file date. If they are still dated 2005, 2006, and 2007, rename them from .sys to .old. When you reboot, they will not load. This will break whatever device or software the drivers are associated with.
Once you know your drivers are good, only hardware is left. We know that the dead battery is not the problem; but we don't know about your converter. We don't know if the power getting to your computer is correct. Can you borrow another converter, suitable for your laptop, and try it? A faulty power supply can defintely trigger BSOD's.
I recommend that you test your RAM with Memtest86 following the instructions in this tutorial: RAM - Test with Memtest86+. Run Memtest from a cold boot and let it run for at least seven passes. This will 4-6 hours. You don't have monitor it. If you should see errors, you can stop the test. Post back with your results.