Try ditching the scales, and instead, play the arpeggio of the chords in the song. Scales always sound mechanical, but the chord tones are much more musical and add some color to the music. If you don't know arpeggios, learn them. if you learn JUST the 7 chords for instance (major, minor, dominant, diminished) you can do so much with that and it will open new doors.
Agree - generally structure around chord tones on strong(er) beats with scale notes connecting them together on weak(er) beats. Remember there although there are 4 beats in a bar of 4/4, the first and third are strong and 2 and 4 are weak. 1 is stronger than 3 with 2 and 4 of similar strength. In 3/4 there are three of which 1 is strong beat but in 6/8 there are only 2 beats. If your chords are changing on weak beats (imagine three beats of D, one of A then a bar of G) the harmony is in a kind of limbo - the bass might try to ignore the A and link the previous chord D with the next 'strong' G (e.g. (1) |D-D-D-C#|B-A-G-A| or (2) |D-D-A-Bb|B-A-G-E|) rather than following the changes directly. Remember that it is the bass that creates chord inversions. In the simple example I've given, you end up in (1) with the A and G chords in first inversion (3rd in the bass) with the bass continuing with a diatonic descent to G and in (2) there is a chromatic pull up into the G, again in third inversion then back stepwise to root position. Constantly playing the root in the bass keeps everything in root position regardless of the voicing of other instruments - this can sound crude.
Rhythm is as important - the bass can be as effective at sustaining pulse as it can at disrupting it. Consider suspension, anticipation or retardation (close relatives of suspension) by carrying a chord tone into the next chord before moving to a chord tone on a weaker beat or playing a chord tone from the next chord on the weak beat at the end of the preceding bar. Weak beats are your opportunity to create movement and tension - strong beats are where you resolve the tension, but you don't have to do so on every strong beat - try sustaining the tension through a whole round of changes until the key chord reappears. Also in the rhythmic arena, rests are as important as notes to creating strong rhythms - ending notes on time and playing rests to their full value are as much a part of playing bass as starting the notes on time.
Chromatics and half-tone movement in general are far more 'user friendly' to bass than they are to higher register voices. Simple decorations and ornaments such as strategically placed upper and lower mordents, acciaccatura and appoggiatura can bring even simple lines to life by creating harmonic and rhythmic interest without tramping over the structure or turning your bass line into a bad solo by running scales for the sake of it.
One other thing is register. Sometimes the same line an octave higher is all that is required to add energy. I'll show my age here and cite 'The Logical Song' by Supertramp. The bass part over the instrumental solo is pretty much the same as the bass line through the verses but played an octave higher, but it doesn't just jump the octave. There is a 2-bar segue from the verse into the break which is a diminished arpeggio (the chord is A°) followed by a chromatic push (A-C-Eb-Gb-Bb-B-C') into the solo. Simple but effective and totally logical!
As always, IMHO and YMMV.