I have two recommendations, the second one might be more important than the first one, but I apply both and in that same order.
1 - Use the option that moves the cursor to the closest “zero”. I don’t know how is that option called on the English version of Audacity.
Try to make all your cuts in some of those points.
2 - Use fade out at the end of the part that fits at left of the joint you will make.
Use fade in at the beginning of the part that fits at the right of the joint you will make.
Fading doesn’t need to be long, a faded overlap of a fraction of second might work, you might also try to make the fade out at left to end after the beginning of the fade in at right.