What can I do at my level to fix a toxic environment?

You may have already tried, but the first route I'd take is to consult, buddy-up and try to solve at lease SOME of the issues at you and your peer's level. Fraternization was NOT a big deal when I was in the Army, unlike my Navy days. It was a bit of a shock but somehow everything gelled except with one unit's Provost Marshal whose leadership was abysmal ... no, it was non-existent.
Anyhow, if you can get your peers on board, even one or two, an work to end the fraternization which in today's Army breeds conflicts and issues such as perceived favortism. Failing that, you can help YOUR subordinates by being as fair as possible, as firm as practical and BACK THEM UP when they have legitimate issues. My mantra was, "Shit rolls downhill ... and my job is to spread it evenly and give everyone an equal bucket full." It was also discarding a lot of shit, never letting it soil my troops or affect their morale. If someone has concerns that are really not an issue, but just his/her perception of things, try to convince them of what's actual fact versus BS.
Stand up for your soldiers ... that alone goes a long way for the moral of YOUR subordinates. Hopefully, other NCO's will take notice that you can raise the morale of your soldiers and follow your lead. It's very tough as a buck sergeant, especially if your immediate supervisor is part of the problem with fraternizing and morale issues. You'll probably encounter that one or two subordinates whose relationship with your supervisor undermines your efforts. Follow Regs and unit SOP's to the letter and DOCUMENT everything that you can. Still, if your supervisor is part of the problem, you'll have to work all the harder to get him/her on board.
Finally, don't encourage anyone going to the IG, but IF they do, be prepared to do the right thing either in supporting them, or refuting their claims. Remember, the IG's primary purpose is to review complaints and determie if their chain of command FOLLOWS REGULATIONS.

There are a few things to point out.

1) Your job is *not* to stop a Soldier from going to IG. Remember that the IG's purpose is to make sure that regs are being followed. That's it. If you have a shitty Commander, but they are shitty BY THE BOOKS, IG does nothing. If the Commander is NOT going by the books (e.g. violating profiles) then this will be pointed out for correction - to the Commander AND his/her boss. No one should fear IG. Your job, as far as IG goes is to give Soldiers free access, and to make sure YOU are not hemmed up, by always doing the right thing.

2) Be the change you wish to see. If you want Soldiers to be treated with respect, do it. If you want Soldiers' morale to go up, do team building exercises. Etc. In a company of horribly unmotivated Soldiers, the team that is MILDLY motivated are superstars. And they will soon realize it and feel it, which will start a positive spiral where they become more motivated, making them stand out more, motivating them more, etc.

3) Serenity prayer: accept the things you cannot change, change the things you can, and know the difference. As a SGT, what you can fix is small. But fix what you can. Help your team understand this, too. As a junior Soldier and even a junior leader, one of my biggest frustrations was my inability to fix problems that were out of my sphere of influence. I was banging my head against a brick wall and pissed off that I was having headaches. Understand which walls are brick (useless to fight), which are glass (can be broken through, but not without substantial risk) and which have a door (fully within your ability to fix). If you understand this, and can help your Soldiers understand this, it will *not* magically make the shitty stuff less shitty. But it *will* allow you to focus on solveable problems to make *those* leas shitty, which is still an overall improvement.

4) Remember that everyone has a boss. And almost all of those bosses have an open door policy. Use it wisely and sparingly. But do not be afraid to use it if it is the only way to get results.

5) Personal story time: I got two article 15s as a young Soldier. The first one saw me re-assigned to the A&T Platoon. Technically Ammo and Trans, but commonly refered to as Ass and Trash. Mostly fat-bodies, PT failures, and UCMJs. We had one dynamic NCO who refused to let us be Ass and Trash, made us meet standards, and, most importantly, GAVE A FUCK. He "fixed" many a "broken" Soldier. Shortly after I left that Platoon, fully "rehabilitated" and back on the line, I was the Division Soldier of the Year. And I went on to make SFC, do 20 years, and retire. Never underestimate how much of a difference one good NCO can make in a world of shit.

6) Personal story time, part 2: After that, I got my stripes, became a team leader, and then a squad leader. My squad was the misfits of the Company. The ones who did *not* want to go get drunk at the strip club every weekend or own a personal arsenal or spend every waking moment at the gym. I mean, it was hard to even call them Infantrymen, really. But they had a squad leader who trained them hard, believed in them, and, most importantly, focused on who they were and what they *could* do, rather than what they *didn't* do. When the BN ran a squad competition, we were the top organic squad in the BN (we came in 2nd, because two platoons combined and picked the top 8 to compete as their "squad" out of their pool of about 55). If you can find and capitalize on strengths, even a bunch of misfits can shine.

Not sure what would and would not work. When I was part of HHC 1st Bde, 3AD 76-78, while not the same thing, from a racial point of view, we all got along, spent time together after hours. In 78, I went to CSC 2/32, and it was a different world. The whites, blacks, hispanics all hung together, and there was little interaction after hours between the races. I assume there was history with that, and new people coming in just did what those there already did? I would assume you just try to do the right thing for a period of time and hope over time you can change the culture?

Lead by example.

There is a way to fix it. It’s by demanding that the people around live up to certain values. If you ignore the little things they will get bigger. Check out my podcast and try to apply them to your circumstances. Look into hordan Peterson as well, he has some good answers. Subscribe and help spread the word.
https://m.youtube.com/watch?v=81TZ8vCFv0o