Whirlpool washer tub noise on spin cycle

Hi. I have a Whirlpool direct drive washer (LSR7233EQ0) that makes a noise on spin that sounds like "zzzzp, zzzzp, zzzzp", which gets louder as it spins faster, then a loud "CLUNK", then it starts the zzzp zzzzp noise again. The top of the agitator moves about a half- inch side-to-side while it's spinning - i.e., the path of it's spinning inscribes a circle about a half-inch larger in diameter than it is (does that make sense)? If I physically hold the top of the agitator, allowing it to spin, but not wobble, the noise all but disappears (and there's no CLUNK every so often).
Sooooo, I think it's probably something in the tub's coupling to the drive shaft, the drive block, perhaps? Does that sound like a likely culprit?
I need the spanner tool to get the tub off, right? Is there any way to substitute for that? I tired a hammer and stubby flat-blade screwdriver, but it seems to be on there pretty tight. I hate to spend $20 on a tool I'll likely only use once (anyone want to buy a slightly used spanner tool?). At least it's still a whole lot cheaper than paying someone else to do it for me!
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If your washer has an auger on top of the agitator, and if the auger is what is making noise, then changing the agitator dogs inside the auger (under the cap or fabric softener dispenser, if present) will probably quiet it down. No, you don't need the spanner tool for that repair. Dog problems usually show up in agitation, not spin though. In agatitation, the worn dogs will not grip the auger, so that you won't get ratcheting action. I can't imagine noise coming from the auger in spin though.
One test to try would be to remove the agitator and then put the machine into spin mode. Does it make that noise you spoke of? I am going to guess here, that this is a somewhat uncommon gearcase problem with stripped gears inside. That's a wild guess. You would almost have to look at the clutch to see what it is doing. When in spin, if the clutch is making some wild jerking motions, then the gearcase needs to be replaced. You don't need the spanner for this either.
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I pulled the agitator off, and it still made that sound. When it's spinning, I can grab the top of the agitator and stop it entirely, and the tub continues spinning happily and quietly. I don't think it's a problem with the agitator, but the fact that by doing that, I'm keeping the tub spinning on a relatively stable axis. Letting it go, the spinning axis vibrates around.
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