Video production at higher quality - alternatives and solutions
First variant:
You can try to set resloution of your display to 1920x1080, but I think that you will have scroll on your screen.
I know that with digital connect via DVI port on Windows PC with XP it was real. In the last time I didn't take any experiments.

Second variant:
You can buy FullHD monitor and don't have any problems with this. They are won't cost a lot of money, especially which sell as used.

Third variant:
I don't see what image see your viewers, but I can think, that problem can be in bitrate of image which won't allow to stream quality image. If your viewers view your stream in original quality and they are have problems with reading text, I think that you need to do bigger scale of your desktop or bigger scale of text in your coding software.

Can you show me the settings of OBS stream, screenshot from your PC and screenshot from your viewer?
Also it can be helpful if you will do recording of your screen for your viewers instead of streaming.