I use the long exercises from Syncopation (pp. 37-44, old edition). If your reading isn't together enough to do the long exercises, you can also do this with pp. 10-11, or 29-31, or the one-line syncopation exercises on pp. 33-36. But move to the long exercises as soon as you can.
We're playing in 2/2— cut time. The melody line written in the book (the stems-up part) is predominantly your bass drum part. Ignore the stems-down part in the book. We're going to add 8th notes on the hihat, and do two different things with the snare drum.
First play the entire exercise on the bass drum, add 8th notes on the hihat, and play the snare drum on 3. If there is a rest or a held note on 3, play the snare drum anyway. Don't play the bass drum on 3. So this well-known couple of lines from Reed:
Would be played like this:
The second thing is more involved. This time we're going to play the book rhythm exactly, with the snare drum on 3, or the closest note to it, if there's a rest or held note on 3— the backbeat is displaced. Often the & of 2 sounds best, if it's in the part, but you can try some different things and see what you like.
You could play the whole drill that way, but I do one more thing with it: every two bars I play the entire last half of the measure on the snare drum— all of beats 3 and 4. Or if, because of a rest or tie, the snare is played before 3, you play the whole rest of the measure on the snare, starting on that note. I'm giving detailed instructions, but you're free to do it however you like. Here is that same two lines played that way:
The only weird part is bar 6: there's a held note on 3, so we play the snare on the & of 2, and go ahead and play the rest of the measure on the snare, since it's one of those measures.
Here's another example, the first two lines of Exercise 3— this one has more tied notes and rests on beat 3:
The first way, with the snare drum on every beat 3:
The second way: snare drum on 3 or closest note to it, last half on snare every two measures:
Playing long exercises 1-8 this way makes a decent-length workout. I recommend playing everything at an even, strong volume; the hihat can be lighter, but don't accent it. Advanced students like to play a lot of internal dynamics, accenting the hihat, ghosting things, but in real world playing, playing everything strong = playing effectively. That should be your foundation, at least.
This Betty Davis practice loop @ half note = 64 bpm is excellent for this drill— it's an easy tempo and you'll definitely acquire the intended feel and attitude. Your tempo goal for this should be around half note = mid-90s bpm. If you're really pushing yourself you could get into the low 100s, but at a certain point the cymbal rhythm starts sounding a little silly— at that point I would try a different hihat rhythm.