What am I doing wrong
You are not posting the schematic.
From the pictures it does not look like you have a large smoothing capacitor after the bridge rectifiers.
You do know that the two supplies will not be interdependent.
You have two bridge rectifiers but not enough wires going to them.
the Arduino behaves in a weird way, like going on and off repeatedly,
Sounds like you have the wiring wrong and the regulator is shutting down due to too much current. Then as it shuts down there is no current so the over current protection comes off and it turns on again. This is known technically as hiccuping.