What is "not much"?. For a typical aerofoil drag coefficient increases quite a bit with angle of attack (assuming constant speed) at typical angles of attack flown at cruise, even your own graph shows it. By the way - heavier load hits your pocket book, someone calculated that in typical airline flying a heavy male passenger (117 kg) costs airline about 65% more in fuel than an average 73 kg passenger - again cost attributed to extra drag.
I said "at cruise speed" for a reason. And I was thinking more C172 than B737.
"At cruise speed" 'typically' means a CL around 0.3 to 0.5.
The drag curve is 'typically' designed to be fairly flat for reasonable weights - like the graph I showed.
Not everything is 'typical', of course
