It is not possible to see what Arduino you are using.
God designed the Uno with a regulator and appropriate capacitors.
The original 7805 was unstable with the wrong input and output capacitors.
Providing you have an appropriate current limiting resistor on the 16x2 backlight, the regulator will not have a high power dissipation.
My suggestion that raw PWM might cause electrical noise interference does not apply for A0. It is beig used as Digital pin.
Seriously, the 16x2 should be stable if you have sensible wiring layout.
Please post a photo of your pcb and soldering.
David.