Power supply issues when using LiquidCrystal and WS2811
The WS2811, in individual led mode, where each ws2811 controls only one RGB led, takes 18~20mA per channel. Three channels (RGB) times 18mA. Times 29 that you are using. That is 3 * 0.018 * 29 = 1.566 Amps at full draw (all leds full on, i.e. White). You are using a 12v input to the Arduino Nano's 5v Linear Regulator, a TI UA78M05CDCYRG3. It is a 500mA regulator.

In all cases, you are overloading both the 5v regulator, and the 1 Amp power supply, as well as wasting alot of that energy in heat. The reason that the LCD is flickering, is because while the WS2811 is fairly wide range and can deal with voltage droop (Voltage drop from high current draw), the LCD cannot. Anything lower than 4.8 or 4.9 volts, and you lose power to the LCD segment controllers, the contrast control, and possibly the backlight. The LCD takes hardly any current itself, but its backlight might need a few hundred mA.

Solution, use a switching regulator with the 12v 1A (12w) supply to get 5v 2A (10W) which will cover both the 29 leds at full 18~20mA and the LCD + Backlight + Arduino's power consumption.

Alternatively, use a bigger 12v 2.5A supply, a separate linear regulator able to handle 1.75A for the led strip.