I went for a ride yesterday, too. It wasn't too bad until I turned west. Then it got chilly. Still, it wasn't a bad day for a ride. I've ridden about 70 miles this January, which is 70 more than I've ever ridden this time of year. Usually, it's just too cold and wet, snowy, or icy.
As far as taking your hand of the bar, I would suggest making sure that you're not gripping the bar too tightly. Relax your upper body and keep your gaze well ahead of you. Steadiness is also a function of your pedal stroke. If you're mashing (trying to push too hard of a gear), it will make your whole body herky jerky. Spin in a relatively easy (but not too easy) of a gear and try to spin in complete circles.
Beyond that, it just takes practice. Just keep doing what your doing, Catrin. You'll get there!
Live with intention. Walk to the edge. Listen hard. Practice wellness. Play with abandon. Laugh. Choose with no regret. Continue to learn. Appreciate your friends. Do what you love. Live as if this is all there is.
--Mary Anne Radmacher