Does the lm317TG require a load resistor in the final design or just for testing?
If the component is LM317TG from OnSemi, it says typically at least of 3.5mA current draw is needed to stay within regulation, but it can be as much as 10mA.
Your resistor network does not guarantee this.
Basically, it makes very little sense to try using so large resistances between Vout and Vadj to try conserve idle power consumption, and then add an extra load to stay within regulation. Adding the extra load on output is also pretty useless, as you need to make sure at least 10mA is always drawn, so 10mA at lowest voltage means much higher current at higher voltages.
It would be best to follow the datasheet, and use a single fixed resistor between Vout and Vadj to make sure it always consumes 5mA, or maybe 10mA even.