You must keep in mind that much of the popular music of the period sounds "classical" or even tends towards "operatta" (this was, after all, after Gilbert and Sullivan had made their mark). Thus, you may have more luck finding what you want in the category of 1910-pop. (Even early ragtime pieces were played somewhat conservatively, since the music form was so new.)
May I suggest for your listening pleasure the following CD: "Titanic: Music as Heard on the at Fateful Night", featuring Ian Whitcomb and the White Star Orchestra, available from Rhino Records on the internet.
It contains many, many popular songs of the day, played just as one might have heard them played on the ship (strictly as background/mood music). At least a few have that "near-classical" sound, yet, surprisingly, came from a hit Broadway or British music hall show.
There's also another CD which may be harder to find but here it is: "Music from the Titanic: 21 Authentic Songs From the Epic Journey". This CD has an almost identical song roster as the previously-mentioned CD, only this time around the music is accompanied by voices singing the lyrics (apparently verboten on Titanic).
All I can say is: in the era before Freud, some of those lyrics...