Heres a slight upgrade
I had the same problem with an emachine model T3958 failing, this has stock intel motherboard D845GVSR and the bestec250 W supply.
At first I figured I would just replace the mobo and supply, it had worked fine a couple years so I figured to keep chancing it, even with the design issue. As you found it is limited supply at retailers online, I found one, but at the cost, I started to compare costs on - a supported Mobo, new processor.
If you look at intel site, as well the chipset is not supported anymore. The next generation is largely supported now, using the 945 series. There is some differences, improvements which cause upgrades of components.
I went with changing over the processor to the 775 socket. I stayed with the uAtx form factor, which ended up being very compatible. The motherboard I wound up with (there are a good number of different feature balance, similar cost range if you constrain to "775" celeron D and uAtx) is the Intel D945GCPE.
I used the processor celeron D 352, which is 3.33 Mhz. I had to upgrade the ram to using DDR2, so used 1GB PC2-5300, 667 MHz.
I selected a little more supply than I think I need, so Antec EA430. Im pretty sure the 350s still work fine in going to the 945 features (more speed, more juice), but this gives me a little more system headroom (for instance, this Celeron is said to be very overclockable) as well as running cooler, quieter and more reliably. 20 dollars more or so, I wanted to be where Id never replace it again, and with the Antec reliability and quality, I believe Im there.
Another thing I did is to go to a vista compatible Hitachi Deskstar 160 GB Sata 2. The new mobo only offers a single EIDE connector, so right now the old drive is kludged in(not mounted just sitting there so I must be careful) as slave in the 5inch bay (the old topology of CD/HD in different bay locations precludes any design reasonable solution of having both drives on one cable.
The motherboard has a lot of Sata 2 support, and I read the newer emachines in this series are using this as well, so thier bios's should keep having support. Plus now Im vista ready, although I found after the install CD, booting into windows that the activation of the OS was fairly easy by getting a new code from Windows.
The costs of components are near the same as the new emachines with these features, but I wont be seeing another bestec this way. It is about the same cost as the D945 motherboard being sold by online retailers, which look like its just marked up end of life stock (primarily corporate users etc, who have tax and capital expense rules that make them have to do funny things - from the price points It doesnt look like emachines/ intel are trying to steer us this way).
Lets see , roughly ~$260 +/- shipping.
works fine though.