Which camera - Sony A7 III vs. Sony A6400

I have a Sony A7iii - it used to be my main camera for a few years, then became my backup when I got an A1. I don't have an A6400, but I do have it's ancient relative, the Nex 7.

Both cameras will take great photos. Don't worry too much about the low light advantage of full frame - it's not significant, and modern noise-reduction software works miracles. Image Stabilisation is another matter. While Sony cameras of that vintage were never class-leading in their stabe, having IBIS does help. I know some lenses have build in IS, and that's better than nothing. But if you're planning on shooing a lot of handheld at low shutter speeds, it's definitely something to consider.

Ergonomics is the main differentiator between FF and APS-C cameras, and that can run both ways. While Sony's FF cameras aren't that massive to begin with (in comparison to older DSLRs or even Canon and Nikon's offerings), they won't fit in a jacket pocket (lens attached). APS-C gives you a smaller body and smaller lenses, which some people find very attractive. The best camera is always the one you have with you, and if you feel your camera kit is too big and cumbersome, you won't bring it out. The downside to the smaller body is that button and dial availability is more limited, and some features are compromised (EVF size, second SD slot, battery size). Sony's firmware has great scope for customising the function of buttons and dials, but that is enhanced by the A7iii having more controls on the body.

I always hate it when someone asks "A or B", and an idiot like me answers "C", but have you considered the A6600? I don't know how it's price compares to your budget or expectations, but it has IBIS, a better EVF and a bigger battery (the same battery as the Sony FF cameras) with a better grip than the A6400 (without the body being too big), so it goes some way to mitigate against some of the traditional compromises that crop body's have had.

Full frame lenses are also (in general) more expensive, bigger and heavier than their APS-C equivalents.

In summary, I think the advantage of FF is mostly in the ergonomics of shooting - having the extra buttons, having the versatility of a second card slot, the better balance of the bigger body in your hand, the position of the EVF in the centre of the camera - things like that that don't show up on the spec sheets. APS-C gives you relatively better pocketability and a better price. I think image quality wise, you can't go wrong with either.