Most people use thicker paper (anything from 120-200 gsm, about 1.5-2.5x thicker as regular printer paper) so the paper model will be stronger.
I always use 200 gsm and never photopaper or glossy paper or any other paper with a special coating or anything, although other people say they do use that without any problems.
I always use a inkjet printer because that's the one I have, but some people also say that the laserjet toner particles can crack and chip when you fold the paper. But again, other people say they don't have that problem at all...
Do you have the same problem with your regular printer paper with the ink still smudging even after a long time? Maybe you can try to print a sheet of paper on a different printer at your friend's home or at school/work, to see if it's the paper of the printer?
I think it's best not to use paper that's supersmooth, or that has a special, smooth coating. Maybe you can explain your problems at the store where you bought your paper, and see if they can recommend other paper and maybe even let you try a few sheets to see if it works better?
I hope you can find some paper that will work better!