I have a 1984 Nishiki International whose front wheel starts to oscillate if I ride no-handed leaning back on the seat. But not leaning forward. I have been through multiple tires and truings, But your situation sounds a lot more severe as I can shift and change hand positions without twitch.

I would check the wheels, both, for straightness (truing). Also, check to see the wheels are in alignment. Before getting lower gears fitted I would have to ride out of the saddle more on hills, and all that mashing would shift the front of the rear wheel to the non-drive side, which seemed to aggravate the twitching.

Also I had a broken rear axel for a few years before the gear change last spring. You would think I'd notice, but it worked okay. The result was that the wheel would go out of alignment a little more often. Now it stays in alignment better, but even when all it aligned and trued, there is still a little twitch when riding no-handed, but not as bad as when the rear wheel would shift out of alignment under heavy mashing.

One last thing...I am not a bicycle mechanic by a long shot, but is the frame or fork out of alignment or slightly bent? My International has a light, thin steel frame which makes it very flexible, which makes it a fantastically comfortable ride, but I sometimes wonder about it bending from road shocks or worse.