On the chassis similar to the OP's that we had, there seemed to be separate flashers. Parts lookups indicate that too. We know where the Turn Signal Flasher is, corner of the Fuse Panel. OP can confirm by pulling it out and seeing if the Four-Ways still work. If they still do, then clearly they have their own flasher. Might be possible to listen under the dash and hear it click. The flashers were one of the few things we never tinkered with on that '83 E350. Replaced some lamps, but that was it, the nearly 10 years we owned it.
In some vehicles, there's a spring clip under the dash that holds the top of the Flasher. In a vehicle that old, I'd expect the Flasher to be out of the Clip, dangling someplace. That makes it quieter since it doesn't have the Dash to reverb on.
It's time to get busy with one of these:
Far easier to check automotive circuits than dinking around with a Meter.