It depends on the steering of your bike. Mostly I think it has to do with wheel flop. It’s not a matter of headset angle or trail alone (although these combine together to produce flop).

Flop is the tendency for the wheel to turn because of gravity. It makes it easier to steer by shifting your weight, but makes the bike less stable. On my Bianchi with low wheel flop I can easily ride no hands, pedal, and even corner. I can even shake my hips back and forth it’s so stable. On my Schwinn with massive wheel flop, I have to concentrate to ride no hands just going straight.

Unless you have a really nice bike, or are exceptionally athletic, you probably cannot sit up slowly. Too much weight is on your front wheel and it wants to turn. You have to push yourself off the bars (carefully) and sit straight up. The further back your weight is, the easier it is. Sit on the back of your saddle or lean your shoulders back a bit.