Actually, we don't often use our 12V systems to drive resistors!
In fact the current drawn by a motor (such as a starter or windlass) CAN go up as the volts drop, because it will be much closer to being stalled and doesn't generate the back emf. It's no longer running at such an efficient point.
Also as it runs slower, it has to run for longer....so more heat. Particularly in the windings of the motor itself.
Also things like inverters can operate as constant power devices, the regulation in them can cause the input current drawn to rise as the volts drop.
Ohm's Law only applies in ohmic systems, the original has the phrase 'metals at a constant temperature' in there somewhere.
However, lunchtime digressions over, Marsupial makes a very valid point about looking after the connections as well as sizing the cable.