It doesn't surprise me that being nearly two inches short, the toilet won't fit!
Just how 'short' is it when trying to install the new one? You can fudge maybe as much as 1/4-3/8", but not any more by having the toilet slightly off-center on the flange.
Industry standard toilets in the USA have been 12" for ages. Whenever you get to 10 or 14" (the only other ones available), your choices go way down, and sometimes, the prices go way up since they don't make as many.
So, your best choice may be to move the flange to 12" to expand the toilet choices. Your 10" one will fit, but it will stick out into the room those extra 2". Personally, if I were going to go to the trouble of moving it (and it sounds like your choice is keep the old one or move it), I'd go to 12", and not 10.