shuts down when asleep, randomly asks to shut down

Hello all,

My 2nd gen macbook (with Core Duo processor and 2GB 667MHz DDR2 SDRAM) running Snow Leopard (10.6.8) has been showing the following two symptoms:

1) When it goes to sleep (whether after the preset time or I shut the clamshell), after about two seconds it will shut down (with a little "pew" sound)

2) If I leave it on for about 1-2 minutes, a dialogue box pops up asking if I want to sleep, hybernate, or shut down the computer (or cancel). If I press cancel, it stays awake. If I'm using the computer, nothing happens. But if I get up to get a drink or whatever, the box will pop up. If I fail to respond within a few seconds, the computer shuts down.

This computer was in the generation where the heat sinks were mis-soldered to the motherboard, causing overheating and random shutdowns, but that was fixed long ago (I received this computer new in 2006), but those incidents were entirely different from what's going on here. The SMC firmware update (1.1) was installed when it came out, and I've confirmed that it's installed.

I have done the following:

1) extended Apple Hardware Test, which found no issues

2) reset the SMC using the shut down - remove battery - press "power" for at least 5s - install battery - turn on

3) reset the PRAM, which fixed the issue for about 8 hours and now it has recurred.

4) run several permissions corrections

5) run a hard disk verification (which showed no problems)

My battery is currently at 631 cycles and 98% health according to iStat Pro. These symptoms show regardless of being plugged in, plugged in without the battery installed, and regardless of battery charge. This is not the original battery.

Being a student, I'm trying to decide if this is an issue which requires expert attention (and an expert bill), or if I need to start selling plasma to purchase a replacement. Any insight would be helpful (note: I'm no computer engineer, so easy on the acronyms and jargon, please :-). Thank you!