Counting is a lot easier to learn if you have someone else to count out loud with you. This way, you can copy the other person in real time.
It’s not necessary or advisable to count all of your playing at full speed. And if you start that way, you’re certainly going to be frustrated. Often, we can play much faster than we can speak. Counting aloud is a tool for learning, getting better command of new material and rhythms, and keeping track through the occasional tough section of a chart. But it can also really deepen your sense of timing, and allow you to learn new things faster — IF you put some time into developing counting as its own skill first.
Someone mentioned counting the 16ths aloud while playing beats and fills, and this is a VERY worthwhile exercise. Of course, you’ll need to simplify your drumming considerably at first. You mentioned that you sometimes have trouble counting ahead of a fill. Take a simple 4/4 beat, with a full of two 16th notes (the “and ah”) abs count it as you play. Then add a 16th note to the fill, and start it on the “e”. Continue adding 16th notes to the fill, one at a time, until you have practiced fills starting at every subdivision in the measure. It may take a month or two until you’re improvising and playing and counting things nearer the apex of your abilities.
At the same time, grab a copy of Syncopation and — before you play a single note — start counting the exercises. The accent exercises are especially good. To practice counting in 6/8, you can use the triplet accent pages. Just use your voice and the metronome for a few days. Another great book for this purpose is The Modern Reading Text in 4/4 by Louie Bellson (no 6/8 exercises though).
In addition, I think there are some good 6/8 materials in A Fresh Approach to the Snare Drum — can someone confirm this?
In essence, you’ll want to develop your ability to count as a separate skill, at the very beginning. Then add the hands. Finally, you can move on to counting while playing the whole kit.