+ knudsenuk
Your approach of changing NPN transistors to PNP transistors and reversing diodes is correct.
But as mvs says in reply #1, the LM337 is not an exact compliment of the LM317.
One important area is frequency stability. The LM317 does not need an input or output capacitor. The LM337 does not require an input capacitor either but it does require an output capacitor. Although both regulators require an input capacitor if they have more than 12 inches of wire (6inches feed and 6 inches return) between their input and the raw power supply reservoir capacitor. But, it is good practice to fit input and output capacitors to both devices, and this is the approach used in most applications.
The LM317 high voltage circuit, described in National LB47, can't use an input capacitor because that would cause excess voltage across the LM317 if the LM317 output were shorted. An input capacitor may cause other problems, I suspect.
So the bottom line is that the more tolerant LM317 is stable in the HV circuit, whereas the less tolerant LM337 is not. As a result the LM337 is oscillating, which would give all sorts of weird behaviors.
Of course, this is just a theory and the problem may lie elsewhere, but you could try connecting a 1uf low ESR capacitor from the output of the LM337 to the 0V line.
You could also try connecting a capacitor with the same characteristics from the input of the LM337 to the 0V line.
CORRECTION 2018_11_01 (see following posts about suitable capacitors)
Suitable capacitors would be 2uF, or higher, X7R dielectric ceramic types or 1uF, or higher, metal film polypropylene types.
Be interesting to hear what happens
Datasheets:
LM317
http://pdf.datasheetcatalog.com/datasheet/nationalsemiconductor/DS009063.PDFLM337
http://www.ti.com/lit/ds/symlink/lm137.pdf