With the 79 series a little more output capacitance helps a lot, and here is where you use a plain old non low ESR electrolytic, in addition to the tantalum, around 220uF 25v works well, has enough capacitance to work, and a high enough ESR to damp oscillations nicely. Unlike the positive ones, the negative ones really want to be power oscillators, and need extra damping that you need to provide, though if you are using a low ESR capacitor just put in a series resistor, any value from 0.1R to 1R in series with the capacitor will do. The 100n ceramic caps right next to the pins are also good, you never know from batch to batch and manufacturer to manufacturer which ones are on the edge stability wise with them.
As to the 723, yes it will work, and you can get really low dropout with a PNP pass transistor, or even use it to make a really high power ideal shunt zener diode. I have even used them as voltage references, the internal one is quite reasonable for the age of the design, provided that you pay attention to power dissipation in the device.