What you seek, my friend, is "the groove". As you're discovering, there's more to it than the mechanical (or even mathematical) approach of playing certain notes with a triplet rhythm. While your approach is technically correct, I'm guessing it lacks the feeling you're looking for. That's what's known as "groove" or being "in the pocket" or (especially for drummers) the backbeat.
There's a subtle laziness that's required in order to really get that feeling of a swing, shuffle, or backbeat. With bass players, it's especially tricky because you need to both push the groove (by being dead on for certain rhythmic parts) but also get a bit lazy (in order to really capture that swing feel). Try thinking about playing the shuffle notes just a tad bit late. Even some of the downbeats can be a bit behind the beat. Your drummer is probably nailing the groove, so let him/her do it. Hit beats 1 and 3 dead on. Everything else can be just the slightest little bit late. Try to avoid being perfectly mathematical about it. Just let those other notes land a bit later.
Listen to some of the jazz and funk greats and you'll start to hear the groove. Early rock-n-rollers also "got" it. It's fun when you get used to hearing it on those old records (early Elvis, Fats Domino, Bill Hailey, etc) because some guys in the band weren't groovin', while other guys were really swinging it. The clash of styles really accentuates the rhythm. Also, all of those funk drum beats that hip-hop artists steal are all excellent examples of totally nailing the groove. (P-funk, the Meters, James Brown, early EWF, early Kool and the Gang, early Commodores, etc).
So many folks have a hard time describing what the groove really is. (Michael Jackson tried to explain it to his band as "getting outta bed on a saturday morning".) That's why it can be elusive. It's tough to explain, tougher to master. But once you hear it, you can't not hear it.