Should my stairs have less than 10 inch run?
Code dictates a tread depth of no less than 10". The rise should be no more than 7 3/4", but since your house had an original stair system that predated anything like that they require now, "grandfathered" is the term that comes to mind. If your structure, as in walls, floors etc., keep your steps from being brought up to code, there is an exception to the code that you can go to.

With that in mind, do as you plan, add the extra tread and increase the depth of tread to 10", that is a close as you can get to code without moving walls etc.

You will appreciate the difference...