Controlling DC Motor with L293

Looking at that motor wiring diagram, it appears that you have no option but to use the limit switches. In that case, for bidirectional control you need to add two diodes, as done here:

The idea is that when one switch opens up at the end of travel, the diode allows you to reverse the current flow so that the motor can be backed away from that end. You will have to experiment to see which switches are activated, and which way to place the diodes (they MUST oppose each other).

Most likely you will use pins 1 and 6 as the power leads, and connect the diodes to some combination of 3,4, and 5. And what are the "auxiliary limit switches"?

As it is, I don't completely believe the diagram posted previously and below, as the NO (normally open) terminal associated with switch 3 is shown as closed.
mot1.jpg