I was tinkering with that new board (in the top-right on photo), adding components semi-randomly, while probing with oscilloscope, and one thing unexpectedly made the system work as was intended at the first place. All what needed was one more diode drop from transcoder board to driver circuits. The display system now looks ugly, with wires everywhere, but it works now!
There is other problem had been uncovered though -- when the 3rd bit of the second digit is set (x1xx, displays '4' when all x are zeroes), all other digits have 4 added to them. Hope to solve this new problem soon, and then the whole system could be nicely packed in enclosure and display will be complete.