When properly executed, the Running Weave should not give you a sense of urgency about "getting around" your partner, any more than any other basic turn. This feeling is usually indicative of either a weak or nonexistent swing, or in the case of promenade, poor tracking.
The beginning of a Running Weave from PP should feel almost identical to the beginning of a basic Natural Turn from PP, except for the timing. You should drive through on the right leg on 1, then begin your upswing as your body passes over the foot. The upswing should continue as though you were dancing a Natural Turn right up to the point the left foot strikes. At this point you will have only partially closed to your partner.
When weight begins to transfer the left foot, you begin your transition into the next shape, and this is where it parts company with the Natural Turn. During this "downswing" action, your knee should soften (although you won't actually lower completely to your heel), you will dissolve the rightward shape, and you will finish closing the position.
Steps 3-4 are similar to 2-3 of a Back Feather. As you begin to move away from the left foot, you begin the next upswing. There will be some body rotation in order to open the right side and allow the lady to swing outside partner on the following step. However, this is moderate body rotation only, and should be complete by the time the right foot strikes. The majority of shape comes from the upswing, not the rotation. All of this technique is very similar to the man's part of Back Feather (or lady's part of a standard Feather Step, minus the toe release).
An urgency to get around your partner probably means you're not swinging up to cause the closing of position; You're just pivoting around her. It's not impossible to dance a Running Weave this way, but it's not much different than dancing a basic Natural Turn without swing... Quite forced and lacking in both shape and volume.
Tracking is also an issue. Make sure you're not cutting in front of her path on the initial driving step, in an effort to make your rotation easier. When you cut your partner off, she can't drive forward, and it loses the linear momentum that allows for a good upswing, which in turn facilitates the rotation. This concept is counterintuitive to most beginners.
I recommend you practice steps 1-2 very slowly, the upswing portion only, to the point your left foot strikes, but stopping before any real weight begins moving into the left foot (the downswing portion). Pay very close attention to the tracking on 1, making sure she is able to step forward easily with good tracking, a heel lead, and no smaller than your step. Pay close attention to the inclination of your left side as you move between right and left, and although you will rotate to some extent, try to keep the left leg swinging in a "forwardly" direction. That is to say, the actual direction will change as you rotate, but not so much that it becomes straight sideways or worse, backwards.
When you get really familiar with the proper mechanics of the entry, the rest usually takes care of itself. Or so I have found with my students.
My preferred timing is 1&23 because I like to draw out the second upswing. However, in your case you might want to stick to 12&3 for now, because the more you concentrate on making steps 1-2 feel exactly like a Natural Turn from PP, the better. Once you've mastered the movement, you might be able to adjust your timing without the sense of urgency that is currently giving you trouble.
Regards,
Jonathan Atkinson
www.ballroomdancers.com