Looking over songs that were popular after the 1992 peak in the violence rate -- as the culture became a lot less energetic, carefree, and emotionally open -- some stand out for being catchy rather than repulsive or saccharine. They sound like they came from a slightly earlier time, and in fact they did.
During the 1991-92 period, one phase of the zeitgeist was grinding to a halt and the society was switching gears to a new one. To make the picture clearer, let's just look at songs that hit it big in 1993 or later, after the limbo period.
"Two Princes" by Spin Doctors was recorded back in 1990 (and released in '91), a year whose culture-feel is indistinguishable from the late 1980s.
"Hey Jealousy" and "Found Out About You" by Gin Blossoms were originally recorded and released in 1989, during the heyday of college rock. When they were re-recorded for their 1992 album, the tempo was a little slower and the mood a little more reflective, but they hardly sound any more different than a song's studio and live versions performed in the same year.
"I'm Gonna Be (500 Miles)" by The Proclaimers became famous when it was later included on the Benny & Joon soundtrack, but it was first recorded and released for their 1988 album.
"Show Me Love" by Robin S. was recorded in 1990, although it was given somewhat different instrumentation in the version that became a hit. What carries the song is her voice, though, and the vocal track sounds the same. What a breath of fresh air when dance vocals had moved toward expressionless speaking instead of singing -- C+C Music Factory, Real McCoy, Crystal Waters, etc.
"Can't Help Falling in Love" by UB40 was a cover of an Elvis song from 1961, even if the slightly faster reggae interpretation did make it sound fairly different.
"Come Undone" by Duran Duran is a bit more of an exception, although the English homicide rate didn't peak until 1995, so it was still made in the creators' rising-crime environment.
"Dreamlover" by Mariah Carey gets some recognition for not having been recorded before its 1993 release, although how catchy it is can be debated. It's a bit too consciously sweet for the listener to get lost in it and feel like singing it the rest of the day. That's about as good of an exception as I could find.
This handful of catchy tunes were all popular in '93 and '94, and after that you couldn't even find a handful, aside from the 2003-'06 period where they consciously tried copying the sound of the late '70s through the mid-'80s. In one way or another, the seeming exceptions after 1992 are rooted in a rising-crime zeitgeist.