You would get better resolution using a binary- or logarithmic-value resistor set; eg. each resistor something like 1.5 or 2x (2/3 or half) the value of the ones either side of it. Also add one another step higher than the highest switched one, permanently connected.
Switching combinations of those in then give you a wide range of fine changes, rather than just a few divisions of 270 Ohms.
Using eight outputs and eight scaled resistors plus the permanent one give 256 unique values.
eg. 10, 22, 39, 82, 150, 270, 470, 1K + 1k8 permanent should give a decent range, if not absolutely linear.
edit - ps. Looking at your circuit again; if you need more than eight extra output bits, just feed the next shift register input from the serial output of the preceding one (pin 9 for the 595), and parallel the clock & latch pins.
Then send 16 (or 24 or however many bits) before sensing the latch pulse to transfer everything to the outputs.