Unstable ESP8266 - flickering lights
The VIN pin on the NodeMCU board can take the 5V directly. Had no problem with that way until now.

What is strange to me is that the current from your 5V to 3V regulator  is degrading. The regulator on the board it's self is basically the same (maybe another manufacturer). What may be the cause is that the regulator is overheating and somehow that the effects the output current and/or voltage.

What you can also try is a LM2596 step down regulator. 
Those are pretty cheap as well and have a good output capacitance. They can deliver up to 3A at down to ~1.3V. Input voltage is somewhere between 40V and your output voltage plus 1 or 2 volts to let it properly regulate.
You can use that module to transform your 5V down to 3.3V easily - i've done it so often with it, i lost count.