Generally speaking, we determine which definition of a multiple-meaning word the speaker/writer meant by the context. This, of course, can be used for humour:
The town drunk got up off the park bench, stumbled down the street, and walked into a bar. "Ouch!" he said.
This is equally true of the word daddy. Yes, it can have sexual overtones. The Washington Post quotes the Random House Dictionary of American Slang saying that the notion of daddy meaning a pimp goes back to the late 1600s and in Blues songs going back to the early 1900s.
Esquire magazine makes the claim that "no one over the age of 12 is calling anyone "daddy" without a tinge of something else there," something I disagree with completely. (They also suggest that the pimp definition dates to 1821.) They go on to discuss various meanings as well as other forms of the word.
The website Whimn discusses "Why Some Women Love Calling Their Partner ‘Daddy’ In Bed" which basically comes down to feeling safe enough to give up control to someone else and enjoy being submissive in a consentual, trusting situation.
So, yes, there are many cases where the word Daddy has sexual overtones and, in our hypersexualized society, the media definitely uses that to boost ratings.
But there are also many who don't see things that way. Recently, a woman posted online saying that her ex-husband had told their kids (ages 10, 9, and 7.5) that they were too old to call him Daddy -- despite the ex still calling his own father Daddy. Reactions were mixed, of course, with some saying you're never too old to call your father daddy and others saying the opposite.
So, when I hear someone, especially a child call their father Daddy, there's nothing sexual there. If a woman calls her husband daddy, it might be sexual and it might not be. It really depends on the context.
As to why someone would avoid the use of Daddy, other than family culture (my parents were always Mom and Dad; I have no idea why), well, yes, there could be some confusion, but I don't think the notion of censoring oneself from using Daddy is all that common, especially among parents.
Is the primary usage of “Daddy” sexual now?
In some circles, yes, others, no. In general, I don't think so, unless context dictates otherwise.
Update: I polled my three kids this morning on the way to school and they all said they would use Dad. When I asked about alternative meanings for Daddy, my just-started-high-school, puberty-induced-dirty-minded daughter came up with the sexual connotations, though she admitted she didn't understand it (and thought it gross). So maybe the definition varies based on whether or not someone has gone through puberty?