You can practice difficult finger movements, let’s call them finger steps, on the saxophone just like you do with complex dance steps. Repeat a certain tricky sequence several times in a row. Start slowly, and gradually increase the pace. It’s more easy and more fun if you play it in a loop. If it goes well, you switch to legato. You can clearly hear when your fingers are not properly timed yet at opening and closing the valves. Slowly increase the speed, and you notice that you are almost not thinking about it anymore. You create a motor memory for that phrase in your fingers. The moment you encounter that passage in the piece you are practicing, you will notice that you now get through it much more smoothly.
Dancers practice before large mirrors and can clearly see when their moves are done more fluently and relaxed. With the saxophone or other horns, you can’t see the moves with your eyes; you ‘look’ with your attention. You get more and more aware of your instrument and the connection you are developing with it.
Tip:
With a long, difficult passage, you start at the end with a not too long part. If that goes well, you start a bit earlier in that part. This way you slowly practice the entire passage. Have fun with it!