Hey GG,
I coach a lot of sports and this related advice might help:

1) Keep your head still.
Racquet sports require this to focus on the object of attention: the ball.
So..

2)focus your eyes on a specific object on the road ahead, like a line, or rock that you want to avoid, on the vanishing point if you like. Find a focus point that works for you.
To shoot a basketball accurately you must keep the head still and focus on the target, the same principle should work for cycling.

3) Lastly, that's all about treating the symptoms, but what is the cause?
Possibly, you are trying to drive too big a gear. When TdF riders start rocking and rolling it is because they are tired, going uphill, and the lowest gear they have is still too big. So try riding in a lower gear and spinning. The cycle stroke, if performed well, should stabilize your upper body.

your fellow Clyde,
PoC

Well done, that's a terrific time!
What's your secret?

PoC

My secret for the 3:35 marathon time is twofold, one I run with a group on Saturdays of FAST guys, most of them have run under 3:30, and a couple have run under 3 hours. And two, I ride with a fast group on Wednesday nights. On most of the fast rides, I end up getting dropped after 15 or 20 miles, but since last year I've gotten much better. I used to get dropped in the first 2 or 3 miles. The area we ride in runs into and out of a gorge multiple times, so every ride has 5 or 6 really steep climbs.

As for riding rollers, I like the idea of setting it up in a doorway, but unfortunately for me, all of my doorways seem to have stairs associated with them, or sharp corners and breakable things!