Yup, I have these songs too, and I know exactly what you mean. Try a lighter touch on those rolled chords, loosen your wrists and let them flow. It’s easy to tense up and get choppy in these sections. If you have to, drop the four finger chords and play them with three fingers instead. There’s no shame in being comfortable with the song, and it’s bound to sound better if you are. Assess, adapt, and overcome! 
Happy to help in any way, Anna. Helping in and of itself is thanks enough.
Karen