Voltage problems, cutting out when hot

A ballast resistor creates voltage drop. So 12 or 13 in = 7 or 8 out. Your pics show 10.X in and less than 6 out to coil. will your coil function properly on 6 volts? In theory, if you're loosing voltage between the battery and ignition system, the charging system is suffering low voltage too. In most case both are fed by the same blue wire from ignition switch. So 10.X at resistor would be 10.X at voltage regulator also.