The channels have a feedback loop, what you see on the display is a read back of what the output actually is, not what it is set to, so from that you know that the unit is getting the correct feedback, so therefore it must be an output control issue.
The output is fed via a mosfet, which is controlled by a couple of op-amps and also ties to the supervisor IC (at least for channel 2), which is used to shut off the output at power up to give time for things to stabilise, my initial suspicion would be the supervisor IC going leaky and starting to turn off the mosfet, you can test voltages at the mosfet feed circuitry to confirm (it is passed through a diode as is the control loop so you can see which signal path is pulling it down, the control loop or the supervisor IC).
Channel 1 has an op amp connected in a similar way to the supervisor IC of Channel 2, test each channel independently as they are slightly different, but linked in some ways.
I cover this circuitry a bit in the repair videos I did on a unit, which explains how that section works. I have a few videos on the unit so you may be interested in them.
I discuss the circuit design at 35 minutes on the Part 2 video.
Full playlist for all E3646A videos I have done:
https://www.youtube.com/playlist?list=PL2z3V9RkHQE2Twk6VoEBURx88TDXPKVSP