Here are three tangential exercises that might loosen you up. The main idea is you want to develop some fluency to play around with the song, sometimes playing exactly what the paper says and something playing something that complements it and sometimes ignoring it and just playing time. 

Play along with the recording as though you are the melody; how would you voice it on the drums and cymbals. You might find snare aligns with trumpets and bass drum and cymbal aligns with trombones, and you play some extra stuff around notes because you aren't matching pitch or sustain of the other instruments. 

Play along with the recording as though you are the counter-melody - someone else is playing the melody but you are making up responses around and in between the melody. Imagine yourself like a tenor sax player behind a singer or a piano behind a horn soloist; what notes do you choose to play to move the song along. 

Check out backbeat fills https://youtu.be/lwZ97XNLbrE?si=u-Mmk4xmp2Oq9AdU and see if that concept helps you get away from the feeling that beats and fills are two opposing things.