Okay, that gives us a few things to look at.
First, and simplest, it's possible that the displays are drawing more current than the supply can deliver. An individual LED only draws about 20mA, but they add up. In this case, you've got 64 LEDs in the each of the 8x8 displays, plus 32 LEDs in each of the 4x7-segment displays, for a total of 192 devices that can draw power. Your peak draw would be 3.84 amps plus whatever it takes to keep the support circuits running. If we assume 50% of the LEDs are off at any given moment, the nominal load would still be around 2A.
The second option is closely related to that, but is more of a wiring issue than a supply issue. Every wire and connector has some small (or not so small) amount of resistance. When you run currents in the Amp range through those resistances, you get voltage.. for the 2A nominal value mentioned above, 1 ohm of resistance in your wiring will produce 2v of voltage drop.
If you have the power connections wired in series, each display sees the combined voltage drop of all the ones between it and the power supply. If the display that's flickering is the one farthest out in the chain, that's probably where the trouble is.
To avoid those problems, use a 'star' topology for your power connections: give each display its own set of wires directly to the power supply.
If that doesn't work (or if you've already done all that and are still getting flicker), we'll probably need to look at the I2C connections.
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