not quite sure aboutyour BIOS upgrade here pal, as i may have similar issues myself as well with my old (A08) BIOS, but i may be able to help you with the quick / unexpected shutdown of your system in other ways since i'm having more or less a similar problem mysef:
a. how much memory do you have installed?
b. how many external / internal devices (HD units, memory stick, speaker systems, cellphone chargers, etc) are connected to your laptop on a regular basis?
you surely know that the more such items (high amount of memory, storage units etc) are connected to a computer system and IF they are draining power from that system alone (instead of getting power from their own external PSUs) that would mean the PSU goes bust sooner than expected.
also having old batteries that cannot charge up fully anymore connceted to the same system for long times, also affects PSU performance badly, as may have been your case as well as my own case with a bad PSU surely.
b. have you 'serviced' your laptop?
if so, when? and if you did, did you CAREFULLY remove all extra dust, dirt and debris from the inside of the laptop AS WELL AS from the memory sockets and anywhere 'contacts' are present, including (but not limited to) the power adapter jack / plug (on both the computer as well as the power adapter itself) and so forth? (that also includes the two or three prongs jack that goes into the wall outlet btw!)
you have to use rather strong blower (duster spray cans for example) for that purpose BUT don't use it directly on sensitive components though: blow the dust at an angle while making sure the dust goes away (in open air but where there is no rain or other 'dagner' involoved) and doesn't come back to sit on the motherboard or other devices again. and then you have to follow the blow up cleaining with cotton swaps soaked in alcohol to clean up all contacts on the memory module sockets, the memory modules 'feet' or 'teeth' themselves as well as the USB sockets / jacks etc and so forth.
then start running the computer again and see if you still get the same quick shutdown as before.
if so, make sure your memory module(s) are also healthy by running tests (such as MemTest86 etc.) and then if these memory modules are faulty, or any other component is faulty (by running diagnosis software that checks all system parts and everything) you have to take care of those items first before trying any kind of upgrade, including the BIOS. (it is best however to clean up and check contacts / connections for intergrity first before running such tests though to make sure you don't have 'physical' problems on the system and it is truly hardware or software issues that are harming it.)
my computer still works fine enough with the old A08 BIOS albeit lately the PSU has gone almost bust (as wel as the battery doesn't charge up anymore either!)
it is now running at a lower speed (600MHz instead of the boosted 1.3GHz) which also caused me to take one of the 512MB memory modules out because the fatigued PSU cannnot handle 1GB memory anymore, otherwise, everything is working fine enough.
of course i'll have to get a new PSU as well as a new battery too, but am also considering buying a new / modern laptop if i have to spend some extra money on just 'refurbishing' this old laptop, which has probably seen its days. after all the whole thing may not be worth it anymore as laptops do not have a very long life cycle anyway.
all n all, make sure your system is not suffering from other issues other than the old BIOS before updating that.
updating to the latest bios in this case might n ot fix the problem...
I have the similar problem and will try to downgrade bios to A06 in order to avoid Dell's update with A07 and later bios that has to do something with thermal issues and they changed their thermal algorithms to basically half the processing power when the RAM reaches a certain temperature... and that also might turn off your computer too...
So as the prveious post:
1. Clean your laptop inside out... not just with air, but you have to open it and take the dust out...
2. your power supply might not be giving enough juice, you said you replaced it... if it is not genuine dell you might always run into the problem
3. you battery is dead and can't get charged... i have one like that in front of me right now...
4. try to replace bios... however you might run into the problems since bios update requires battery to be present in laptop before it proceeds with update...