Some of the stuff that people recommended are actually late-comer bands and not what I'd call the real influential classics of the genre. Most bands that put out their "main" LPs after '95-'97 could hardly be called "classic", IMHO. There were a lot of crap bands during that time too. 90s metalcore is probably one of the most misunderstood subgenres in hardcore simply because the stuff that became popular in the 2000s overshadowed a lot of the good stuff. Then you wind up with a bunch of people repeating stuff on the internet that weren't actually there at the time.
Anyway, here are what I'd consider some "good" or "important" LPs. Note that I'm only going to mention a few things from the 80s...
Cro-Mags: first two LPs are essential, with the first (Age of Quarrel) being the much better of the 2 IMHO.
Agnostic Front: Live at CBGB - this is great because the recording is pretty good and it acts a nice "greatest hits" type overview of AF in both their crossover and more pure HC eras.
Sick of it All: first 7'' and first LP are the most essential. Albums 2 and 3 from the 90s are also worth hearing. Scratch the Surface was a big favorite of mine when it came out and when I saw them on the tour.
Note that while I love a lot of old NYHC (the aforementioned, Warzone, Antidote, etc.), I cannot stand later bands like Madball and such, which is why you won't find them on my 90s list.
Starkweather: into the wire (Crossbearer, which is really a comp. of demos that got put out as an LP came first and was arguable more influential but I like this one a lot more because the recording and performance are better IMHO...haven't really delved into their more recent material.)
Unbroken: life.love.regret and final 2 7''s(One of the most important albums to usher in the era of chunky hardcore and of course, Slayer worship. Unfortunately inspired a lot of crap bands, but this LP is a classic).
108: creation sustenance 2XCD (Pretty much everything they recorded was good, but my favorite was the final pre-reunion LP. Yes, they were Hare Krishnas but you won't care about that when you hear them.)
Integrity: first 2 LPs (Probably THE band responsible for popularizing "evil" metal-type themes in HC.)
Mayday: everything you can find (Very influential in the early 90s, unfortunately never released an LP or got good distribution so a lot of people have never heard of them. Wish someone would put out a discog CD.)
Bloodlet: eclectic CD (comp. of early 7''s which was their best/most original material. First LP is pretty good but not as important)
Groundwork: discog CD (Kinda like Unbroken, but a lot more political, raw and unhinged sounding)
Acme: discog CD (Sounds like Rorshach covering Slayer with better recording and playing than the former)
Mean Season: discog CD (The Grace LP is really good, but the vox are kinda whiny and weird so it's something of a grower. But once you get into it, it's great stuff. )
Converge: now, I don't think this band has done much that's impressive in over a decade, but their first couple LPs and the Unloved and Weeded Out sessions are classics. Third LP is enjoyable, but by the time Jane Doe came around I was losing interest. If you listen closely, you'll basically see that they were trying to combine aspects of Rorshach, Starkweather, and maybe Acme and Voivod. They did a good job at it too.
Ringworm: never been a huge fan of this band but they're worth mentioning. Stick with the early stuff.
Catharsis: everything they did is good, but the Samsara LP is their real classic. I was lucky enough to see them on an early tour before they recorded this one.
Cave In: until your hear stops LP (Probably the last truly great 90s metalcore LP to be released.)
Coalesce: nowadays people go crazy over the later stuff but for those of us paying attention early on, they were already kinda "done" by the time the first LP came out. The s/t 7'' and the Earache version of the 002 e.p. (NOT the rerecording) are classics.
Refused: songs to fan the flames (I know it's cool to hate this band because they're popular, but this is a great LP...you get chunky riffs, some Slayer worship, and even a few hints of Fugazi along with the political lyrics of course)
Lash Out: there is a discog thing that is already sold out/OOP, but this band did some really great stuff in the 90s. I like the "What Absence Yields" LP and "Worn Path" 12'' a lot.
His Hero is Gone: yeah they were a "crusty" band, but all the metalcore kids were listening to them back in the 90s. First LP is more grindy, second is probably a little more aligned with metalcore.
Uranus/Union of Uranus: the 2x7'' was great, kinda muddy and screamy metallic HC with loads of atmosphere...believe there is a discog CD now.
One Eyed God Prophecy: LP - this was like Uranus, but with a more drawn-out Neurosis, vibe. I think they shared members also, but I'm not 100% sure on that.
Threadbare: feeling older faster e.p. (similar to 108 but more structured, and lyrics more on the personal tip rather than Krishna)
Rorschach: I think there are a couple different discog CDs now. I much prefer the first LP, which reminds me Black Flag a bit, over the second one. I might be in the minority though.
These are just off the top of my head. I'm sure I'm leaving a couple things out but these are what spring to mind when I think of essential recordings from the 90s metalcore era.