Ben’s answer is quite good, and the bottom line, that slashing damage isn’t going to work on brown puddings, is spot-on.
He is also completely correct about mindlessness not being a defense here. Mindlessness only protects against spells that try to get into your head (with the mind-affecting tag). Most illusions don’t; most illusions actually produce a real, physical effect, whether that be light or sounds or whatever else. A mindless creature will experience those lights or sounds exactly the same way it experiences the lights or sounds of real objects, and being mindless is probably even more susceptible to being fooled since they cannot reason that something cannot be real.
But what I really wanted to address is blind-but-blindsight. Illusion spells can create sounds, vibrations (which are just sounds by another name anyway), smells, and so on. Which means it is entirely plausible for “a nonvisual sense (or a combination of such senses) [used] to operate effectively without vision” to be fooled by an Illusion spell.
Moreover, Illusion (Shadow) spells are partially real, created out of the shadow-stuff of the Shadow Plane. It’s no different from wall of stone conjuring stone out of the Plane of Earth, only it gets classified as part of the Illusion school instead of the Conjuration school because... it does. That classification makes sense to arcane scholars in the world; maybe the Plane of Shadow has some connection to illusory magics, maybe it just happens to be taught as part of the standard illusion curriculum rather than a standard conjuration curriculum. We don’t know, and it doesn’t really matter to us. The point is, despite being an Illusion, Shadow spells are at least in part real.
Which means whatever “nonvisual sense (or a combination of senses)” the ooze is using, it’s still going to detect the legion of sentinels because the legionnaires are, in fact, actually there to detect. They have volume and probably even mass, there are firm boundaries between the sentinel and the air they exist in, and they produce vibrations and possibly even scents.
In short, an Illusion (Shadow) spell should affect blind-but-blindsight creatures.
That doesn’t change the fact that the sentinels do slashing damage, which isn’t useful here anyway.