We all have dominate sides whether on land or in water. In water it is more about your low side (skating) edge. My fav is my left side edge and I could stay on that edge all day if I could. My right side edge always needs more management and practice to hold the edge clean. I like Coach Dave Cameron's analogy, "chocolate and broccoli" sides. My left side is my chocolate side, but need to force myself to eat more broccoli :-)
However, I'm right handed and my right side recovery hand/arm wants to participate more with added moves that caused some imbalance. Those added moves took some time to remove, but can sneak back in without me knowing it right away; this becomes obvious when getting air off my right shoulder and I find more water than normal being an inch or so lower.
This doesn't answer your question integrating swimming into your other sports, but only that it's ok and normal to have a dominate side (or edge) in the water and to add more practice on your less dominate edge to maintain stroke/edge symmetry. Become aware of your dominate hand/arm and leg doing unnecessary/added movements (above and below surface) which can trigger imbalances as well.