The problem, I think, fundamentally, is that I bought the computer from a kind of skeezy tiny retail shop in town.  I thought if something went wrong, I would have more of a safety net than if I got it on eBay or something.  That assumption proved unfounded, when it turned out they sold me the wrong power adapter, presumably because it was cheaper for them to do so.

If it turns out that the right power supply will let me run at full speed without overheating — I’ll find the money somehow.

Thanks for finding the product spec page!

Thank you, but I know that — that’s what I meant by “This is apparently not the correct one” (that wasn’t meant snarkily.)  I have not ordered the appropriate one for financial reasons.  Indeed, I’ve back-burnered it because the computer works partially at least.  What I don’t want to end up doing is pay for a new ps if it turns out that’s not the problem, and I will still face the overheating.

are you certain the cooling fan is working?

Not to worry: this is precisely the sort of thing I am famous for forgetting!  The cooling fan audibly spins, increases in volume as the temperature of the machine increases, and blows a nonzero amount of hot air.  Whether it’s working optimally or not, I don’t know.

http://www.dell.com/downloads/global/products/latit/en/spec_latit_d810_en.pdf

Secondly, the recommended AC Adapter/Power Supply for a Latitude D810 is 90W (19.5V/4.62A) (similar for C810 model). I cannot say whether 65W supply is your problem, but it seems you aren’t using the correct one.