dying battery, increasing resistance
"so the open circuit voltage increases in time"


That would be the case if the the short-circuit current remained the same, but that decreases as R increases. So that point moves up the i-axis. At the same time the open-circuit voltage (the so-called cell's EMF) decreases as well, moving that point to the left on the v-axis. That's another reason for the lower short-circuit current. The result will be that the slope of the line will become more shallow.  

So a battery is not a fixed voltage with a varying series resistor; the voltage is variable as well.

Added (upon Russell's request)  

Like I said also open-circuit voltage decreases, which has nothing to do with internal resistance; in an open-circuit there's no current, so also no voltage drop across the internal resistance. It's the cell's actual EMF which decreases:


  In practice, cell EMF depends on temperature and concentration of reactants and products.
   If the concentration of reactants increases relative to products, the cell reaction becomes more spontaneous and the EMF increases.
      As the cell operates, the reactants are used up as more product is formed causing the EMF to decrease. 


(from here)