The ADJ pin current (100uA max for both the LM117 and the LM317) is almost nothing and doesn't change. The entire chip's idle current must flow through the output which must be drained to ground through the voltage-setting resistors. The idle current is half as much for the more expensive LM117.
By "idle current" you mean Adjust Pin Current?
The adjust pin current can be from "50 to 100 uA" between different ICs, which amounts up to a 2% error in voltage regulation if a 5 mA (240ohm) programming resistor is used. In practice (i.e. in a mass-produced circuit with fixed resistors) this'll contribute this much error in
initial voltage setting accuracy. Some people think that specifying 0.1 % resistors prevents this occurring, well it doesn't!!
As for adjust pin current variation, both the 117 and the 317's APC can change by 5 uA which amounts to 0.1% variation in output voltage (for a 240 ohm programming resistor). Not a lot, as you say. Though the 117 technically has a greater variation in APC, because of the higher operating temperature range.
But if application is likely to cause the APC to significantly increase over life (such as for spaceflight, where there is ionising radiation degrading hFE), then
decreasing the CPR to below 120 ohms helps a lot.
Anyway.. Isn't the 117/317 both come from the same silicon wafer? I'll guess the cherry-picked ones from the middle of the wafer become classed as 117 and the runner ups become 317s. That'll mean that some "317"s are almost "117"s. Also I wouldn't be suprised if all the H/T/M/etc suffix current rating 117/317s all actually used the exact same chip inside, and the max output current being to do with the amount of metal in the package.
So... there still is the OP problem: why's the output voltage collapsing under 'load'? I think the IC's overloaded myself.....