What was the resolution of the older PC's screen? If it was some knackered old laptop then it might well not have been much higher vertical res than the DS has. If you have a shiny new 1440 vertical res screen or something then it is going to look different.
Equally if you are on a new version of desmume without your previous settings you might be using the stock graphics settings. I do not have a current build in front of me by the one I do have has a whole bunch of options under config for interpolation, ranging from the classic sai and super sai to slightly more exotic offerings. Try those.
You might also wish to try
https://gbatemp.net/threads/hi-resolution-ds-emulation.364549/