Both Sony A7c and Sony A7 III have Full frame sized 24.0 MP resolution sensors so sensor size and resolution is not a differentiator between these two cameras.
Below you can see the A7c and A7 III sensor size comparison.
Sensor Size and Resolution Comparison image of Sony A7c and Sony A7 III Cameras
Sony A7c and Sony A7 III have the same sensor sizes so they will provide same level of control over the depth of field when used with same focal length and aperture.
DxOMark Sensor Scores
DxOMark is a benchmark that scientifically assesses image quality of camera sensors. It scores camera sensors for color depth (DXO Portrait), dynamic range (DXO Landscape) and low-light sensitivity (DXO Sports), and also gives them an overall score. Sony A7c and Sony A7 III sensors have been tested by DxO and the results show that A7 III has a better overall score of 96, 1 points higher compared to A7c's score of 95.
Model
Overall
Color Depth
Dynamic Range
Low-light ISO
Sony A7c
95
25.0 bits
14.7 Evs
3407 ISO
Sony A7 III
96
25.0 bits
14.7 Evs
3730 ISO
Sensor size sets the circle of confusion, so it drives how much depth of field each
of these bodies gives you at a given aperture. You can calculate it exactly for the
Sony A7c
or the
Sony A7 III ,
including hyperfocal distance.