Controlling for ambient temperature changes in a transimpedance amplifier
The temperature coefficient of resistors tends to be pretty low, compared to what (I understand) you are seeing, typically tens or hundreds of PPM (parts per million), but check the datasheet for your resistors to be sure.  Consider using a constant current source to drive your LED, rather than a voltage source through a resistor.  With only 100 ohms I suspect you'll have a pretty strong dependence on supply voltage.
Also check the datasheet for your photodiode.  Some have a much different temperature coefficient when operated in photovoltaic mode rather than by controlling a current (the photodiode will produce some voltage when it sees light).
Your ideas about feedback to control the LED current also has merit, but you might also consider instead making an identical copy of the circuit.  On the copy, don't have the light pass through the translucent material, but just directly illuminate the photodiode.  Connect this circuit to another ADC input and use it as a reference for comparison.  If the components are matched reasonably well, this will cancel out a lot of the temperature and/or voltage dependence you typically see.
Also, the ADS1115 has an internal voltage reference, so the ADC should give very stable readings when presented with a constant voltage.  However, if your supply voltage varies, then any signal derived from your supply voltage also varies.  While this circuit may not benefit from it a great deal, many sensor circuits are ratiometric, providing a signal that is proportional to the power supply voltages.  In that case you want your ADC to use the supply rail as a reference, rather than a fixed voltage.