Christians have engaged in a very old debate that has to do with being in the world without being of the world. I submit that the biblical view makes a very stark distinction. It's not “fuzzy” or ambiguous at all.
Romans 12:2 in the J. B. Phillips translation may be my favorite paraphrased verse of any I have seen: “Don’t let the world around you squeeze you into its own mould, but let God re-mould your minds from within . . .” (RSV: “Do not be conformed to this world but be transformed by the renewal of your mind”).
Any discussion that takes into consideration the above aspects in relation to evangelism must ponder and interpret St. Paul's famous statement of methodology:
1 Corinthians 9:19-23 (RSV) For though I am free from all men, I have made myself a slave to all, that I might win the more.  To the Jews I became as a Jew, in order to win Jews; to those under the law I became as one under the law -- though not being myself under the law -- that I might win those under the law.  To those outside the law I became as one outside the law -- not being without law toward God but under the law of Christ -- that I might win those outside the law.  To the weak I became weak, that I might win the weak. I have become all things to all men, that I might by all means save some.  I do it all for the sake of the gospel, that I may share in its blessings.
Almost all Christians would readily agree that there are obviously limitations to this methodology at some point down the line. Differences arise as to where the line should be drawn. Being “all things to all men” does not, of course, mean sinning in order to win men.
We don't visit a prostitute and employ her services in order to proclaim Catholic truths to her (so she can know that we are “regular folks” and not like all those dime-a-dozen puritanistic, straight-laced, judgmental, anti-sex Christians, and therefore listen to us). That is mortal sin, as we know.
I sarcastically exaggerate in order to to get my point across. But a more apt scenario, which is actually being used now by some very well-known Catholic writers, involves the use of “bad” or “rough” (or – take your pick – dirty, cussing, swearing, obscene, vulgar, profane, “R-rated”) language in order to “better relate” to those who use the same language.
One might surmise that such behavior would come under one of the categories that Paul lists above: perhaps being weak in order to win the weak. That's at least within the realm of possibility or plausibility. But I really think not, because of what Paul teaches so often elsewhere in his letters. He's very concerned about Christlike behavior. I don't see a hint anywhere of him communicating this sort of approach: being rough and tumble in language to those who are the same, in order to win them over.
If people with whom we are sharing the gospel and truths of the Catholic faith see us fall and stumble, then they see that we are human just as they are, and not occupiers of some high, sublime quasi-heaven which is completely apart from the realities of the world and our own concupiscence (“pie-in-the-sky”).
They're bound to see that, anyway, if they're with us more than half a day, so that consideration will “naturally” take care of itself. It doesn't follow, however, that we willfully, deliberately put on more coarse attributes, and enter into near occasions of sin, so that we will be liked by these people.
I don't think that non-believers or those who are wholly secular or nominal in religious matters think that all religious people are hypocrites and sanctimonious jerks. I agree that they may think many are that, and indeed this is what atheist and secular propaganda constantly promotes: unfortunately aided by our frequent sins that are consistent with such an impression.
I think they're perceptive enough to also note examples of people who differ from the world: who stand out. This appeals to them, and makes them wonder why the person is different: what caused it; what makes them tick; why are they so “happy”?
In the final analysis, St. Paul, who repeatedly tells us to imitate him, as he imitates Christ (1 Cor 11:1; cf. 1 Cor 4:16; Phil 3:17; 4:8-9; 1 Thess 1:6-7; 2:10; 2 Thess 3:7-9), is quite clear about how we ought to act. It's no mystery at all.
We're not to be as much like the world and worldly behavior as we can, so the lost and the seeking will think we're cool and identify us as one of them. Rather, he teaches that Christians ought to exemplify a stark contrast with the world:
Ephesians 4:24, 29 and put on the new nature, created after the likeness of God in true righteousness and holiness. . . .  Let no evil talk come out of your mouths, but only such as is good for edifying, as fits the occasion, that it may impart grace to those who hear.
Ephesians 5:4 Let there be no filthiness, nor silly talk, nor levity, which are not fitting; . . .
Philippians 2:15 . . . be blameless and innocent, children of God without blemish in the midst of a crooked and perverse generation, among whom you shine as lights in the world, (cf. Matt 5:14-16)
Colossians 3:8, 12 But now put them all away: anger, wrath, malice, slander, and foul talk from your mouth. . . .  Put on then, as God's chosen ones, holy and beloved, compassion, kindness, lowliness, meekness, and patience,
1 Timothy 4:12 . . . set the believers an example in speech and conduct, in love, in faith, in purity. (cf. 3:10)