Explanation how uln2003a works
I expect the ground symbol is to represent the connection to the IC substrate.

The diodes connected to the input and output are almost certainly parasitic diodes that are not really required but are created as part of the semiconductor design process.  Since they are reverse biased under normal conditions they do not affect operation and the circuit can be analyzed as if they are not there.  Current will not flow through them under the conditions you will be using.

I don't understand your explanation of what is going wrong.  You should be able to drive all the channels independently.

Make sure that the E terminal (pin 8) is connected to the negative terminal of the power supply feeding the Arduino - it is important they share the same ground. If you have a separate power supply for the motor it's negative terminal should also be connected to the E terminal.

The COM terminal should connect to a snubber network, or zener diode to control the back-EMF of the motor to a reasonable value. Connecting it to the positive terminal of the power supply may stop proper action of the stepper motor except at slow speeds.