Can't find any low ohm resistors with decent wattage to put between Vin and the LM2588 Vin...left it out for now...
Got rid of the pots. Using 2x 4.7K resistors for the 'top' of the feedback resistor chain, and a 1K resistor for the 'bottom'. In a perfect world, that should come out to 12.792 volts at the output.
No load results (practically no load...feedback network on the positive rail provides about ~10.4K, added 10K to ground on negative rail, about ~2.5mA idling load)
Vin = 5.35v measured at the LM2588 and coil primary side
Vout+ = 12.62v
Vout- = 12.62v
V @ feedback pin = 1.217v
1.5K load on each rail (3 470 ohm resistors in series, in this case ~19.2mA total load post-regulator)
Vin = 5.34v measured at the LM2588 and coil primary side
Vout+ = 12.51v
Vout- = 12.52v
V @ feedback pin = 1.207v
82 ohm load on each rail (in this case, ~210mA total load post-regulator)
Vin = 5.17v
Vout+ = 8.48v
Vout- = 8.48v
V @ feedback pin = .829v

Put the meter on the ground rails, positive rails, looking for any obvious resistance...
Didn't find anything measurable either in resistance or in voltage differences.

'scope pictures up next if I can get the Holdoff knob to work right and clear up the picture.
(EDITs vs. new posts each time I make 'progress' should keep this a bit cleaner...)
EDIT: I can't get the 'scope pictures to clear up at all. Secondhand Tek 2246A I've had for about 10+ years. Never had it "tuned up" or calibrated other than using the built-in menus.
EDIT#2: Added the "snubber" R & C to the positive output diode. Made it worse. Output voltage dropped down to + and - 8.1v. Noise appeared to stay the same. Removed previously mentioned snubber since all it was snubbing was my power!
EDIT#3: Added a 1uf across LM2588-Vin to ground. Output voltage started at 8.4v, but rose to about 9.1v after the circuit had "warmed up" for about 2 minutes (power resistor load and LM2588 w/heatsinks rose to about 30F above ambient, 105F).
EDIT#4: What if the polarity marking for the coil was actually on pin #1 according to the LM2588 datasheet vs. on pin#2 according to the Coilcraft datasheet? Hmmm....
Edit#4a: Nope that's not it.
EDIT#5: Added a smattering of .01uf caps, one each at the input and output of each positive and negative rail diode, one at the Vin/Pin7 to go along with the 100uf & 1uf, and another at 'top' of the feedback resistor chain. Also put a fat 4700uf cap across the battery input wires on the PCB itself.
Now...I'm getting 11.6v at the output with the 82ohm load...I think. It's seeming like the regulator wants to be "warmed up" before it'll start giving me a decent output.
Using 2 4.7K resistors for the 'top' of the feedback, 1K resistor for the bottom of the feedback.
Vout = 1.23 * ( 1 + R1 / R2 )
12.792 = 1.23 * ( 1 + 9400/1000 )
At any rate, 11.6v is within the tolerances from the resistors in the feedback chain as well as the tolerance of Vref inside the LM2588. The calculated worst case each way gives me a range of 11.453v - 14.294v (5% resistors in the feedback, along with 2% feedback voltage tolerance according to the datasheet).
EDIT#6: No dice...
Loaded each rail down with 44ohms (82ohm + 2x47ohm in parallel). Got 7.4v on each output. That's 168mA @ 7.4v, where I
should be getting ~12v @ ~270mA.

But, after the regulator/heatsink warmed up a bit, 130F (~60F above ambient), I got 9.58v output, with 4.84v on Vin.
If the 70% efficiency I calculated earlier holds up, that's 438mA load at the outputs, 626mA draw from the battery pack.
Rechecked it after the regulator had cooled down to 60F (at the heatsink) and the output was only 7.4v. (Vbatt = 4.82v)
Re-recheck it after the regulator had heated up to about 100F (again at the heatsink), output up to 8.42v. (Vbatt = 4.81v)
Re-Re-rechecked it with the regulator heated up to about 130F, output up to 9.58v. (Vbatt = 4.81v)
That a repeatable response. Did it 3 times, same temps, same voltages, a few millivolts lower on the battery pack each time though.
What the heck is going on here?