First, congratulations on making good progress in only 3 weeks. You sound like you’re well on the way. :0)
So you’re weaving a bit from side to side? That is how unicycles (and bicycles) stay the right way up. A perfect straight line with no weave is (virtually) impossible.
But yes, you do need to keep the weaving to a minimum, for style, and for energy conservation.
First rule: the seat is there to support your weight. second rule: the pedals are there to propel the uni. So, sit solidly on the saddle, and twiddle those pedals. MANY beginners put too much weight on the pedals.
Rule 3: Don’t try to regain a balanced position at the end of each pedal stroke or wheel revolution. Keep moving. SOME beginners ride like this: step, balance, step, balance, step, balance, as if they want to remain in control (or regain control) at the end of every pedal stroke. The proper way (in normal riding!) is to keep moving smoothly, and to make minor adjustments to your speed to keep balanced.
Imagine balancing a broom on your finger (with the head at the top!). You now want to walk across the room. You let the broom start to fall, then you follow it at just the right speed, and you cross the room smoothly. If it falls too quickly, you have to run faster and faster to stop it falling. If you move too quickly, the broom acts like an upside down pendulum, and you frantically move your hand back and forth as you try to regain control.
A unicycle is just the same. The weight falls in one direction. The support (where the tyre touches the ground) moves in the same direction. Lean too far and you have to pedal faster and faster. Pedal too fast and the wheel overtakes the weight, then you have to correct backwards… difficult!
So, in simple terms, aim for smoooooooth. :0)
Suggested ways of developing skills:
First, get used to riding up to, say, 50 pedal strokes.
Then get into the habit of stopping deliberately and dismounting, rather than riding until you fall. The psychological effect is good if you end each ride (however short) with a success (dismount) rather than a failure (fall).
Ride big circles IN EACH DIRECTION. It is tempting to stick to whichever side you find easier, then you can become a ‘one sided’ rider. This is a difficult habit to overcome.
Ride in a confined space. A hall, small carpark (parking lot) or tennis court is ideal. Having a REAL limit to the space available will concentrate your mind and force you to learn to steer. An unlimited straight path or a path round a lake (etc.) will keep drawing you on to covering distance - at the expense of developing skills.
When you can ride a reasonable distance (50 - 100 pedal strokes), steer each way, and dismount comfortably, it’s time to work hard at freemounting - that’s getting on without leaning on something.
When you can freemount, the world is your oyster!
Good luck. :0)