Similar to an Airplane
A sailboat generates power from the interface between the air and water, not just from the presence of wind. If the water were moving in the same direction and at the same speed as the air above, a sailboat would be at rest relative to the air, and wouldn't be able to extract any power from sails. Similarly, a fully airborne craft will be fully "immersed" in the air stream, and will not be able to reliably sail any direction but will instead be carried along in whatever direction the wind is blowing.
However, if you were to attach your anti-ballast to a rope and suspended the entire craft from it, you could reel in and pay out the rope to alternately lift and lower the airship. The CG of the airship/anti-ballast system would not change in the vertical axis (raising the ship would lower the anti-ballast and vice-versa), but it would move the airship relative to the air around it. If you placed sails like an airplane's wings and tail you could glide forward, gaining forward speed while both rising and descending, as well as turn either with a rudder or by banking the entire airship (banking would technically work, but is not a good option for this type of craft, especially with fabric sails, a rigging mistake or gust of wind could violently roll the craft and throw the crew to their deaths).
You would want the tail to be mostly rigid or to employ very taut fabric, and either moved like the rudder of a ship, bent by pulling on lines attached to either side, or equipped with a moving control surface in order to steer. It would be possible to use conventional sail plans, but because they would be completely sideways, the booms would need to be under much more direct control, otherwise all they will wind up facing downward and you will only be able to generate useful thrust while moving upward. It may be easier to build a "wingsail" or something more structurally similar to a fabric and wood windmill blade (think classic Dutch windmills here). These sails or wingsails would need to be able to accommodate wind coming from any direction in the 180 degree forward arc of the ship. A sail would do this by adjusting the lines controlling the sails, while a wing would need to rotate on a pivot aligned with the pitch axis of the airship. You would want to put the axis of the pivot at or ahead of the quarter-chord of the wing(1/4 of the way from the leading edge to the trailing edge of the wing). At the quarter-chord you would experience little to no control forces aside from friction. A pivot ahead of the quarter chord would be somewhat self-tacking and at the very least stable.
The ship would likely start with the anti-ballast fully retracted. the process for a sail based variant will depend on the sail configuration, but for a winged variant, the wings would start pointed mostly down and slightly forward (probably within 10 degrees of straight down). Then the cable would begin paying out rapidly. Wings would generate "lift", but given that they're facing almost straight down, this lift would be thrust relative to the airship. As the forward speed increased, the wings would be slowly angled more and more forward. They would now start to generate an increasing amount of lift, which would slow the descent, but continue to generate thrust. Once the cable was fully payed out, the crew would need to wind it back up. How this is achieved I cannot say without more detailed knowledge of your setting, and it may be the case that the ship climbs too slowly to generate useful thrust. In that case the ship will lurch forward each time the cable pays out, then slowly raise itself back up, then the cycle will repeat. If the crew has some way to reel the cable back in quickly enough, they could angle the wings upward on the return to generate thrust in much the same way they did on the descent. This would also allow them to preserve their momentum from the descent and retain control authority. Note that you don't have to climb all the way back up. The ship could oscillate vertically over the span of just a few feet as long as the crew were quick and accurate enough on the wing/sail adjustments, and the reversals were fast enough (you will generate little to no thrust while reversing direction). Alternatively, with an inherently stable pivot point (forward of the quarter chord) springs or a specially designed steering canard could be used to adjust the angle of attack automatically, making these short pogo hops more practical.