the input ripple at the 7805 is a sine between 6.6V and 8.8V peak to peak which breaks down every 80 ms to about 5 V
??? about the capacitor before the regulator??? In theory 6VAC should gives you about 9 volts near stable depending of your capacitor size. Try increase capacitor value first. Forget the serie diode here.

Reliability and stability of a numeric design sit on the PSU first...