OK, an update:
I connected the external PS to the display board, to supply it stable 5V. No fluctuations of voltage were observed. So, I replaced the 7805 and diode bridge since they could fail under load. Still, the fluctuations are present. Finally, I replaced the 100uF capacitor with a new one, even larger capacitor (1000 uF) and that fixed the fluctuations. Thank you!