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You are right that this resistor is part of the LCD contrast, although it is optional. The LCD contrast pot (on the top side of the board) has one leg connected to GND and the second leg connected the VCC through this resistor. Most LCDs I tried are better off with just the connection to GND, that's why the resistor is missing by default. But you can put a resistor in there if you think that would help, 1Kohms for example. Note that this won't help if your LCD requires a negative VEE, you'll need a charge pump then.

I just tried the default setup again with the 4x16 and it works for me... so I guess the pot has a bad solder or something like that on your particular Dragon...