I have old kohler faucets throughout my house. When I replace a washer, it will not stop leaking until I tighten the nut so tight that is very hard to open/close.

What am I doing wrong?

Have you refaced the seat down inside the valve? Anew washer on a seat that is uneven will not seal. You can buy an inexpensive tool to reface the seat. http://www.homedepot.com/webapp/wcs/stores/servlet/ContentView?pn=Resurfacing_Valve_Seats&langId=-1&storeId=10051&catalogId=10