Yes, you can use sails but they would not look like normal sails and they would work differently.
Basically sails work by ships having lots drag in the width axis this means that the component of wind on that axis can be turned to go backwards. This changes the momentum of the wind on the length axis which creates an equal force to push the ship forwards.
In your case instead of high drag on the width axis, you have functionally infinite drag on the vertical axis. This means that your sails need to work on the vertical component of the wind. That means either the rising or descending air currents.
Birds and gliders make use of such currents, so this is definitely possible. The really difficult part is figuring out what the air currents in your setting would be like. I certainly have no idea. In our world these currents are largely driven by the shapes of the underlying terrain but in your case this would only apply over the islands. Presumably the currents would come from the currents in the underlying "gas giant".
Note that if you can alter the altitude, which you really should, you can easily fake vertical air current by changing altitude.
As for the appearance. The sails would probably be on the sides of the ship, so that the hull does not obstruct the air flow. Alternately or additionally, for stability it might be better to have a catamaran arrangement and you could then have sails between the hulls. Additional sails would be added in front and behind the hull. In any case the ships and sails would extend horizontally instead of upwards. So the ships would be quite flat. With maybe some "castle" structures on the deck for defense and to give officers clear vision of the crew.
The spars the sails attach to would be in the direction of the width axis. They cannot be along the axis you want to move on or the direction the air current you want to use is on, so that is actually the only usable option. Normal ships can use wind from sides or behind but your airships only have usable drag on a single axis so they are more limited. That said if the wind is going in the right direction you'll drift along so you are not actually losing anything apart from rigging complexity.
In fact, you can probably assume that shipping routes will generally be in the direction of the wind and that the actual sails are needed for control and to move sideways to the main air current.
The sails the extend from the spars on the length axis. In practice they would attach to the spar in the forward direction and the "ropes" on their backward edge would attach to the next spar. This would allow the back edge to move in the direction of the air current while the forward edge stays still. This forces the sail into a curved shape that turns the air current backwards and the ship gets forward momentum.
Control would be by trimming the sides differently. The ship will turn away from the side with more sail. In practice the ship would generally be directed 90° from the main wind, since the sails would really only be useful for movement sideways in the wind. Near islands where strong updraughts would exist they would additionally be useful to stop the ship in harbor or to move the ship away from harbor.
Answering the comment by JoeBloggs brought few rather important aspects of this to my mind, so I'll add them here as they are kind of crucial.
Since the ships only move, and only can move, between islands that also float with the same technology and the setting does not have terrain that would make the winds highly variable or unpredictable, your destination would almost always drift along the exact same wind the ship and island of origin drift along. There would be turbulence and storms to mess things up but in general you can ignore the drift along the wind as it does not move the ship significantly relative to the nearby islands.
This obviously makes this type of sailing lot more convenient and practical since the motion the sails extract from vertical air currents would not be just the controllable part of the ships motion, it would probably be the major part of the motion relative to islands and other ships. This would promote the ships from primitive sailing ships to something more similar to early steamships that do not need coal. A big upgrade.
Also, navigation needs to be considered. When your destination is constantly moving with the wind using stars for navigation is not very useful. Even if the winds are very even and predictable the errors would rapidly add up and you wouldn't be able to predict the position of your destination relative to stars for very long or very accurately.
So you'd need to supplement your navigation by observing air currents, birds, and probably spyglasses and lighthouses.