How much contact your finger makes with the fretboard depends on how high the action is, and how high the frets are, so it’s reasonable for there to be contact. I would suggest trying for a gentler touch, and see how it sounds.
How high is the action on your acoustic? Have you had anybody adjust the action or check the neck? You may not be doing anything wrong, it may just be that the truss rod needs adjusting, or the frets are worn.