"Today my partner and I tried hardly to follow anonymous's instructions - get the body physically out of balance - keep the hips with the upper body in a perfect vertical position. It was a great movement but there was one problem coming up. How do you catch your weight really smoothly on the landing foot and how do you get the speed under control?"
It is tricky, isn't it...
Some ideas:
Don't rise very high at first. The less you rise, the smaller your fall will be. Similarly, don't try to bend your knee that much as you do that part of the lowering.
With less rise and fall, you will have less energy in your dancing, and an easier time working out how to sustain the energy. Later on you can increase everything once you have it figured out.
But once the fall is under way, especially once the knee is bending, don't try to hold back. The backwars partner can pace this a little bit, but the forwards one really can't. To control it, carefully time when you start the fall so that you don't get to this point too early. And make sure you are prepared to run with the fall - take all of that energy and draw it out into a nice big action on the drive of the next figure. This energy you carry out of the fall may have you taking those resulting steps much larger than you ever have before, and that's exactly why you move that way - the energy from the fall is what gives you a huge and dramatic drive through the lowered part and into the next rise.
Some teachers also want to break the lowering into two parts while working on control. The first part would be lowering the foot rise - and only as an exercise you could even stop there. The knext part would be bending the standing knee and projecting the body forward. That second part is the one that has to go off balance. Whears the first part will have you off balance but between your feet, so you could always control by weighting your arriving foot sooner.