Now that that's out of the way, let's get on the interesting part. I said that Shriekers are unplayable. And it is because no Strength and no Dexterity means no attack rolls. You are not just bad at attacking, because even being bad at something is very good for the shrieker, you just can't. Which means absolutely no martial class, from monk to warblade, is going to cut it. (I think that it is what it means to have neither of those, since you automatically fail any Str or Dex check. But if I am confused on the rules, please correct me. It would still be extremely bad, but not nearly as much as it is now)
But there are still casters! No. No Strength means no Somatic component, no wielding an arcane or divine focus that you couldn't carry anyway, and no using material components. Clerics and Wizards are not for you either.
Spoiler: Why I didn't include druids
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To be fair, a druid is not that bad, of course, because druids can do anything. You basically play as your animal companion till lv 5, that can carry you around without much problem, then you take Natural Spell, and you spend the rest of your career as whatever broken animal you want. You have -10 Wisdom. Yeah, so what? A druid without spells is still much stronger than most classes. But if you are here, you are here to play a shrieker, not play something else with a huge penalty to mental stats. Also the reason why I didn't include phylactery of change. That would give you physical stats, but these completely defeat the point of trying to assign LA.
What remains? Psionic classes could fit. They don't need somatic components, or focus. However, the shrieker still has a -10 in Intelligence. That is as low as it gets. Even putting all your leveling points in Intelligence, a Shrieker Psion 20 rolling an 18 has 13 in Intelligence. Yep. You are manifesting worse than most level 5 psionic characters. And getting items to boost intelligence will not only get you so far, but you
can't even carry them without a Strength score!! Getting custom ioun stones may be achievable, but so expensive that your character would not manifest anything before 10th level or so.
Then we have the incarnate. Most of its abilities are based of Constitution, so that's good... But remember that all soulmelds must be worn on a useable magic item slot (if you do not bind them, you can wear a magic item on top of it, but you still need the slot). You are a mushroom! No arms, no legs, no head for the crown, no face for the eyes, no chest for the vest, no shoulder and no hands. You may argue for the neck and waist, but that's all. Yeah, melding is out.
The warlocks are known for being able to function with 3s across the board, so surely they would fit, right? Weeeell... No attack roll means no Eldritch Blast (and no glaivelock) until you take Eldritch line at
level 11! And -4 Cha modifier means your invocations with saves will probably never work on anything. It is hard to play a class when two-third of your class features just plain don't work. But you still have other invocations, right? Well, no. Invocations require somatic components, and since you have no Dex, this is also out. Also warlock really don't care much about Constitution. Also, also, even if we handwaved the somatic requirement with some sort of custom feat, a lot of invocations deal with getting you bonus to some skills. Which just won't help you since you don't have the Intelligence to get any skill decently high. There is basically nothing the warlock should do that a shrieker warlock can do. You can't even play a class known to require next to nothing from the character to be played efficiently. That is a testament to the horror that is playing a shrieker.
Binders also rely a lot on Strength and Dexterity, since they are more martial than casters per se.
So what? Are shriekers doomed to only advance by templates and never use a class level effectively? I don't think so. Because if you can shriek, then you may speak. And if you can speak, then you can Truespeak.
No, I'm not even kidding. What do useless HD give you? Maximum skill ranks. What do most utterances not need? Attack rolls and Saves. I believe the optimal class for the shrieker fungus is Truenamer. And when the best class a character can take is Truenamer with a racial -10 to Intelligence, you know there is a big, big problem. (You could say that an ardent with -8 Wis who can't manifest until level 4 would still outperform a Truenamer with only two item slots. And you're probably right. But because the difference between a shrieker and a normal race ardent is way higher than the same for Truenamer, and also because it is still April Fools, we'll go for Truenamer) Boosting a skill is way easier than boosting DC, and even having -5 to all Truespeak check does not amount to a lot when you should have +40 at least to really play a Truenamer. (if you roll at least 16, you have 6 Int, and with 4+int/lv, this is enough to keep Truespeak maxed out and have another skill for prereqs purposes) At least, lots of means to boost Truespeak do not come from items, or not slot items, so there's that. Of course, ask if you can use the Book of Words fix, but even without, you should do a not that much more terrible job than a normal Truenamer.
Obligatory "I have no mouth and I must scream" joke, because that's what you will be doing most of the time if you try playing this.