Hi Mark, and welcome!
You raise an excellent question. Several years ago I got the same advice: “Keep taking the ii-V-Is” as if they were the magical panacea for all ailing jazz pianists. That’s only half the story, though.
As you know from listening (and from your fake books) most every jazz ‘standard’ uses the ii-V-I progression. But, unlike in classical theory, a ii-V does not necessarily mean a real modulation to a new key. Frequently, the Imaj7 chord doesn’t materialize (often becoming another ii chord for the next ii-V).
I love Mark Levine’s ‘Jazz Piano Book’ and agree it’s the bible of voicings (and much else). The problem I found, though, is that he offers numerous voicings but doesn’t say much about learning them except: “Practice them through the cycle in all keys”.
While you should have the circle of fifths down, you don’t have to ‘remember’ every progression as such. What you’re balancing is (a) the shape of a voicing (the way it looks and feels) and (b) the logic behind it (i.e. which notes the voicing includes).
The shape is a visual and a tactile thing. And, if you cycle through keys in a regular order (around the circle, down in whole-tones, etc) it will become familiar through ‘muscle memory’. But that doesn’t help you much when faced with a lead sheet. So you also need to practice chord voicings randomly, as well as other regularly occurring chord patterns than ii-V-I. And on top of all that you need to practice depending on whether you’ll be playing solo, or in a duo, trio or larger combo. Nightmare, eh?
Fortunately, help is at hand! I strongly recommend Phil DeGreg's 'Jazz Keyboard Harmony' (Aebersold Jazz). He systematises a wide variety of basic voicings (guide tones, left-hand, two-handed, fourths, etc) in ii-V-I and several other frequently encountered cycles (ii-V, V7-V7, maj7-maj7, rhythm changes, etc).
Don't be put off by the fact that the book is 'designed for all musicians' (not just pianists). Voicings are a jazz pianist's bread and butter, and I don't think you'll find a better way in than this. What's more, you can continue to apply his techniques to learn any new voicing you come up with (i.e. your own experimentation or using Levine’s ideas).
Another book which I just love is Randy Halberstadt's 'Metaphors for the Musician' (Sher Music). He really opened up Levine (and a bunch of other books) for me. All the well-known method books are essentially giving the same information. Each one just slices the pie a little differently. For me, Mr Halberstadt was the one who made everything start making sense. Was it because I'd already read other books? Hard to say, but his book could stand on its own as a theory/practice handbook. The first edition came out in 2001, which may be why you don't see too many people recommending it.
The most relevant Halberstadt chapter for your question is ‘Voicings 101’ where he offers his take on voicings for specific environments (solo, trio, quartet, etc). It really complements Levine and DeGreg. In fact, every chapter contains a wealth of information presented in a practical and entertaining way.
You might think from all this that I’m some kind of voicing master. Far from it! But I thought my thoughts on ‘getting there’ might help you on your own journey.
Two parting tips:
1) I struggled with the drudgery of learning voicings. So many variations, so many keys! I still struggle but it’s no longer drudgery. Something Kenny Werner wrote made things easier: Start simple and master something basic. Don’t move on until you can do it effortlessly.
He’s right. It’s too easy to fly through things and not really internalize them. I find it helps to think of voicings as a kind of (pseudo)-Zen meditation. Just relax and enjoy seeing they way the chords look and how they feel. Alternatively, while you play through them, really focus on your touch and timing. Play with a metronome if need be. We jazz folk can easily get lost focusing too much on the ‘right notes’, when really good playing is defined the same was as classical playing: tone and time.
2) It’s easy to jump ahead to altering notes in voicings. But master simple rootless voicings with 3rd-5th-7th-9th. You may not use these ‘vanilla voicings’ much in practice, but if you practice them first and really focus on the important 3rd and 7th when you go through them, you’ll find the notes you want to alter later on (9th and 5th/13th) will jump out at you. You’ll play a chord and immediately see ‘ah, there’s the 9th, now it’s become the 13th of the next chord’.
One final thing, then I’m done, I promise

. Although this is a wonderful forum for pianists, you may find your jazz questions answered more comprehensively on allaboutjazz.com. Oh, and you might also think about using a more obvious topic header next time

Have fun,
Goose