Never happened to me something like that, and I used a dozen of 1602 LCDs, most of them I2C, but also directly connected to Arduino (but I2C is the best and current solution for me), and even as shields.
So as you don't seem to have power issues from your Arduino, it sounds like a batch of defective displays to me, especially if you got both from the same reseller (can you tell us which one?).
I'd ask for a refund or at least a replacement, but in any case in the meantime I'd buy another one from a different shop/reseller.