The partnership can get into a wide variety of balance situations in the course of a dance. The more advanced the dancers, the more frequent this situation is likely to occur. The best way to stabilize the partnership is to truly become a two-headed four-legged animal. It seems very logical that if the partnership uses all four legs, it should never be off balance. The solution is to know how to use all four legs.
A four-legged animal is not created by simply standing 2 two-legged animals side-by-side. The two-legged animals are not mirror images of each other. The goal is to create a new entity (a four-legged animal) that did not exist before. This new entity creates feelings that can only be achieved by a four-legged animal. What you feel like when practicing by yourself and what it feels like when you dance in partnership should be very different. The partnership develops four-legged feelings and stability that can not be achieved in any other way.
One of the most obvious ways to be four-legged is to make sure that all four feet are on the floor all of the time. A dancer's feet slide on the floor to maintain contact. Of course there are always some exceptions. The supporting foot has about 98% of the body weight and the so called free foot actually has about 2% of body weight. It really isn't completely free. If the free foot has 2% of the body weight that means that both legs are being used all of the time.
The partnership's frame is the primary physical connection used to create the two-headed four-legged animal. The frame has a forward connection that holds the two partners together in the offset position. This connection should be as light as possible yet strong enough to be able to feel and use your partner. Even though the physical connection is through the frame's arms and hands, the logical connection needs to be between the bodies. When dancing Latin/Rhythm, the body is defined as: "from the bottom of the rib cage and up". When dancing Standard/Smooth, the body is defined as: "from the hip sockets and up". Each partner has a center of gravity and the partnership also has a center of gravity. Because the partners are offset from each other, the common center is diagonally between the two partners. The power for movement comes from the supporting leg and travels through the common center of the partnership. The partnerships center is the reference point for all four-legged movement.
The partner's head position has a big influence on the four-legged animal. There is no "one" head position. There is a general area for the head, but the head is constantly adjusting the balance of the partnership. You see dancers all of the time (especially women) that look like their head is frozen in one position as if they have had a stroke. The head is alive and dynamic, constantly adjusting to the partnership's movement.
It is important that the leader be able to feel the total unit known as the two-headed four-legged animal. The connection has an "equal and opposite" feeling. The connection may change depending on the physicality of the movement, but the connection is still "equal and opposite". Whether you are in a "closed" dance frame or a one hand "open" position, you are still a four-legged animal. Even when you are in a solo (non-connected) position, there is an invisible logical connection that still exists. Make an effort to feel your partner in the "solo" position. It will surprise you. There are many types of invisible connections.
Footwork is also a big factor in stabilizing the four-legged animal. In Waltz and Foxtrot, for example, the first step forward is always a "heel step". The first step provides the power for the movement. This creates the "power and glide" feeling. If the first step forward is a "ball" or "flat" step, the power for the movement is lost and the partner moving forward will disrupt the balance of the partnership. When dancing Waltz, the "rise" and "fall" footwork is critical for balanced movement. The "rise" is accomplished by lifting with the foot (ankle) and leg (knee) at the same time. The "fall" (lowering) is executed by "lowering" through the foot first then "lowering" through the leg second. Lowering through the foot gets the partnership stabilized so a "heel" (power) step can be taken. The "lowering" through the leg initiates the "heel" step. A common problem is trying to take a "heel" step while still on "rise". This is disastrous from a balance standpoint. When the partnership is on "rise", all four feet are still on the floor. This is another common problem, trying to "rise" on just two feet.
Let's talk about "tweezing". "Tweezing" is an action that applies to all of the ballroom dances. Any time a dancer's legs cross; there should not be any air between the tops of the thighs. The very top of the thighs should touch and be tightened even more by a slight rotation of the hips toward the crossed leg. The slight rotation reinforces the crossing connection to make it very solid. In a "5th Position Break" element in American style Rumba, the partner's legs cross behind. The step should feel sideways, not backwards. The legs cross and touch very high on the inside of the thighs, then are solidified by a slight rotation of the hips. We now have a four-legged animal in a very solid position. If I dance a Rumba "Open Break" element, there is air between the thighs because the step moves straight backwards and the legs don't cross.
The more advanced the dancer, the more often you will get into precarious balance situations. Working on four-legged movement in the very beginning of your dancing will reap big rewards in the future.