Popping a 2N5911 dual JFET into the noisy Tektronix AM503 (B064098) eliminated both the noise and the offset problems:

The test signal (yellow) comes from the scope’s calibrator output into a 2320 Ω resistor, so the AM503 calibration is about right: 0.6 mA ≅ 1.5 V/2320 Ω.
Just to maintain historical accuracy in the two AM503 amps in the TM502 mainframe on the Electronics Workbench, I transplanted the good (not noisy) OEM Tek Q230 (from SN B075593) into the previously noisy-and-offset-prone AM503, which now works fine. I now have a pair of works-pretty-good AM503 amps, one not-so-good AM503 in the to-be-fixed lookaside buffer, plus a defunct Q230 dual JFET.
That third amp (B075593, now with the NOS 2N5911) has a nasty noise problem:

The barely visible yellow trace is the same calibrator signal as before, but the output is a howling 4.2 MHz (!) sine wave. The oscillation amplitude responds to the AM503 front panel gain control, making it possible to see what’s going on:

Flipping the front panel switch to limit the AM503 bandwidth to 5 MHz shaves off the fur:

Disconnecting the probe or unplugging P220 kills the oscillation, as does setting the front panel switch to CAL/DC LEVEL, which means it’s an internal feedback problem.
It’s trivially easy to construct an amplifier circuit that becomes an oscillator at the slightest provocation, but this puppy had been working dependably for somebody else during the three decades (!) before I bought it and continued for a few years after that, so the overall circuit topology is known-good.
Shooting this one will require more pondering, as the obvious first step of replacing the power supply’s electrolytic caps had no effect.
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