I've no idea what you are trying to see in that scope shot at 10ns/div. Everything you are seeing is an artifact of how you have set it up and it doesn't indicate anything useful.
The amplifier section should reject almost any conceivable noise in the power supply (OK--not at 100MHz, but....), so if all of the grounds are indeed securely connected together and you have anything reasonable on V+ and V-, the amplifier should not be going all the way negative nor oscillating.
The 120Hz ripple you saw is what you would always see if you overload a full-wave linear power supply like this. The only reason any of us were suggesting looking at the power supply was in case the amplifier mute circuit or some other part was not properly designed or constructed or had failed and was unstable enough that it might be affected by a power supply deviation that would ordinarily not affect a properly set up amplifier. In my case, the only reason to go down that road at all was because you stated that the amplifier had worked properly for a while and then failed in both channels for some unknown reason.
If you have relatively clean and balanced power under no load and small load conditions, but you still have that oscillation or beating, the problem is not the power supply. I would start checking all of the components, but really I think the most likely cause here is that you have somehow blown up your LM3886 chips. They're supposedly well protected and you don't have a lot of voltage, so I'm not sure exactly how that would happen. The big mystery would seem to be why the amplifier feedback circuit does not respond to it going full negative. Since it doesn't do this under no load, it probably isn't a shorted output.
You might start by connecting more channels of your scope right to the LM3886--to both inputs, the output and the mute pin. Seeing those simultaneously (at a slower timebase please!) would be very helpful. Also, if you can, measure R1, R2, R3 and R4 in circuit and check their related capacitors, especially C4.