The second waveform you linked to - with the diodes reattached and with
the resistive loads reapplied - does that include the filtering
circuitry in the schematic, too? (C1561-3, L1562)?

If so, maybe I'm misinterpreting what I'm seeing in the scope picture,
but that semi-square wave with the ringing on the leading edge has an
amplitude of over 17V P-P. That seems excessively high to me for a
rail that's supposed to be delivering +5V DC (especially given it's got
close to 1000 mfd filtering it).

Are you sure that there's not some kind of weird grounding problem
happening here?