Diagonal Patterns of Dots on Backgrounds

My wife has had some problems with my her Windows 98 system (extremely
slow response, e.g, 15-seconds or more between mouse click and next
display). So Santa gave her an eMac for Christmas. Now I'd like to
use it because it is soooo much quiter than my desktop machine.

I tried reinstalling Windows 98, but it crashed, returning to the C
prompt during the final installation phase, right after restarting and
displaying the Windows screen logo.

So I reformatted the drive and tried installing an extra copy of
NT4.40 but it wouldn't install.

Then I tried installing XP Pro, and that went satisfactorily except
now I see on backgrounds (login screen, desktop, etc) diagonal
patterns of dots which disappear once a window is opened and then
closed, again and the background is redrawn. But the dot patterns
reappear when new backgrounds are rendered.

I'm guessing it's a memory problem. The memory test at startup doesn't
report any problems. If it is a memory problem, which memory, the
onboard memory or the 128MB DRAM?

If it is memory, any suggestions for software to testing memory?

If not, what might it be (and the remedy to correct it)?

Thanks,

-Tom