DRO's enhance a machine's accuracy, true. They are independent of any possible lead screw/nut error or wear. However, they have no direct effect on backlash. The issue at hand is the back-driving of the screw from cutting pressure causing pushoff to actually move the entire cross slide. It is likely that wear on the screw and nut are the root causes. That said, adjustment of the gibs, whether they are straight or tapered, should help. In an extreme case, it may not give total relief, but should allow new parts to be made on the machine itself. Trouble with that is you need to disassemble the machine to get all your dimensions, reassemble to machine the parts, then tear it all back down to repair it. Unless you choose to source the parts out of some vendor. I don't know what is available. I imagine most hobbyists would rather pride themselves making them.
In no case should the dials on any machine be used in a bidirectional fashion. As Bill said, backlash is present, and has to be. On NC/CNC machines, they do use bidirectional travel, BUT, they either use an encoder on the stepper motor driving the axis, or a scale that measures actual travel. The computer used has all the backlash comp data stored, so it can offset, or overtravel as needed to maintain accurate positioning. You will see the effect of bad comp data if you program a ball on a lathe, for instance. That requires a cut that reverses direction while forming the profile. A step is discernible almost all the time, because you just about can't keep up with the normal wear on a machine. An old, worn out machine that hasn't been compensated will leave large steps. The same will apply to a mill or machining center if you interpolate a circle.
I remember years ago working with a factory rep on a small mill we had just bought. It would not place holes in a pattern accurately. We took a few test bars and holed them every inch, then every half inch, and plotted them on our CMM. Taking that data, we created an offset profile for X and Y. The machine was locating within 0.0002 throughout the work envelope when we were done.