I remember watching a video that covered experiments proving that antimatter still falls “down” as well, so that avenue isn’t a valid place to look either.
Our understanding right now is that gravity is the shape of spacetime. As the modification of spacetime even on a small scale involves huge masses and thus energy, it doesn't seem possible to modify our local spacetime in a useful way without seriously messing stuff up. It's like removing the Earths gravity, by removing the Earth. If you'd use just a fraction of that energy for a rocket, you'd get a lot further. So unless we are really really wrong about gravity (which we might be, but not likely), anti-gravity is not a thing.
Nobody knows of any such thing as negative mass or anti-gravity, there's no theoretical reason to expect that it exists, and every previous suggestion that something like it had been found was very probably a hoax. But let us know if you find some mysterious rocks that fall upwards in a vacuum.
Full answer: There's some adjacent work being done on the warping of spacetime using the Casimir Effect. If they're able to produce negative energies of sufficient amplitude then such work might eventually lead to anti-gravity applications, but this is considered fringe.
If they're able to produce negative energies of sufficient amplitude
Have we been able to produce any "negative energy" yet? I kind of got the impression that negative energy was some sort of mathematically valid alternate solution to some physics equation, but which didn't have any basis in reality.
Not really. We can make tiny regions of relatively negative energy compared to empty space, but not in a way that does anything more useful than vacuum-welding metals together and there is disagreement as to how far much further down the zero point is.
Scientists who have since studied Brown's devices have not found any anti-gravity effect, and have attributed the noticed motive force to the more well-understood phenomenon of ionic drift or "ion wind" from the air particles, some of which remained even when Brown put his device inside a vacuum chamber. More recent studies at NASA, held at high voltages and proper vacuum conditions, showed no generated force.
Ion thrusters are real (and neat), but they are not anti-gravity tech.