I've found that playing "measure, measure, measure,..." results in the "beat, beat, beat" sound. You have to play "phrasephrasephrase" (<-- no spaces intentionally) to get that flowing movement from the music and even then you have to watch out for the tiniest extra bit of time between the phrases as your mind "resets."
It might help if you think of each phrase as having an invisible slur connecting all the measures, and then another invisible slur connecting all the phrases.
Movement also helps. Play with your body, not just with your fingers/hands/arms.
These are things I'm just now starting to figure out, so take what I say here with a large dose of salt, but slowing way down and trying to play everything in one flowing sequence helps to get rid of the "beat, beat, beat." At least it worked well enough for me to start to play more smoothly. What I did was ensure that when I got to the end of a measure or phrase or wherever I was moving from one sequence to another, I mentally played ahead of my fingers to get the first note of the next sequence integrated with the current sequence. That got rid of the "gap" between sequences. (You can call these "sequences" or measures or phrases or whatever you wish, it's where you play up to a certain point then naturally hang for a fraction of a heartbeat before moving onward.) Slow down and play through those places while concentrating on playing the next note as if it were part of the prior sequence.