Hi Darcy,
As relates primarily to Nikon...
I can't give you great advice on shooting birds in flight as most of the wildlife I shoot is in their natural habitat doing what critters do. Occasionally I'll get a shot off of something in the air and it's actually in focus. But, as a general rule I either stalk critters or wait for them to come to me. I also use a tripod 95% of the time. And bought a gimbal a few years ago to use with it, although that's 50/50 because it's heavy and a hassle IMO. So the gimbal isn't necessary, and just having the support of a tripod can still let you move the camera around while on a stable platform.
My camera settings, regardless of the model, I start with single point AF-C (not AF-S), or 9-point. Make sure under the cameras custom settings menu (A1 and A2 usually) that AF-C priority selection is "release" and AF-S is "focus." Remember though that "release" means the camera will start shooting while it's still aquiring focus, so many will be out of focus, until it locks in. But it's better than having the camera never fire at all! I'll then use manual settings (M) combined with auto-ISO. So I'll set the aperture about as open as it will go, like f/4 or f/5.6, and start with a shutter speed around 1/2500 or so. If critters are slow, you can slow this down to maybe 1/800, or speed up to what works. The slower the better if your getting sharp shots. However, it's pretty much acknowledged these days that the higher mp cameras need fast shutter speeds with wildlife, or sports, or anything fast moving. At least until you develop good tracking skills and consistency. The old rule of 1/focal length no longer applies - you could triple it. And if something pops out of nowhere you want to get the shot, not some blurry mystery object... Then activate auto-ISO and let it do the exposure work, with a ceiling around ISO 1600 or so, depending on the camera (my D8xx cameras I set at ISO 3200). The hardest part with BIF shooting is finding the bird in the viewfinder and then getting an initial lock on the bird, and that's where the lens does make a difference. Things happen fast and you may think it's focused, but it's not. The more points you use, the slower the camera "thinks" which is why in a perfect world single is best, however you need a compromise that works.
Try and position yourself in a way that lights the subject the best, don't just shoot any which way. You want good flattering light on the subject relative to where your shooting from.
Many people are still using the D7000 with it's 16.2 mp sensor and getting perfectly acceptable results, so a camera upgrade might be nice but not as necessary as getting one good lens you use for wildlife. Use that lens for a while, then learn to fine tune it's focus in-camera and practice technique over and over. Truth is, for every one decent picture I get, there are at least 100 that get thrown away for so many reasons - out of focus, bad backgrounds, poor light, etc. etc... the list is endless. As someone once said, just show your best work, not all your practice. Whnever I shoot, it's usually in a burst of 3-5 shots, and the center one is usually the sharpest.
As far as moving from DX to a full frame, many professional wildlife photographers are using DX cameras like the Nikon D7200 or the D500 because of the 1.5 crop factor, which is better than a teleconverter IMO. They'll use full-frame for landscapes. Many have kits of two complimentary cameras. D850 & D500, or D750 and D7200, etc...
If you wanted to blow a small amount of cash today, and that's the problem with good wildlife photography ($$,$$$) --> MONEY!
This is what people are using in the Nikon world (before getting elite):
D500 ($1,900)
or
D7200 ($999)
The D500 is a processional build, very similar to the D850 full frame, where the D7200 is just like your camera, very good build and much lighter. The D500 is also 20.4mp versus the D7200 at 24.2 mp
As far as lenses:
Nikon AF-S 300mm f/4E PF ED VR ($2,000) and carry a Nikon AF-S Teleconverter TC-14E III ($500)
or
Nikon AF-S NIKKOR 200-500mm f/5.6E ED VR ($1,400)
So you can see why people all are buying the 200-500, it's cheaper. The 300mm is the choice when weight and size matters (and it's just a better lens). It's a really small light and sharp lens. It's also a constant aperture.
Other 3rd party choices are the Tamron and Sigma 150-600 lenses. Just be aware that the Sigma "Sport" version is one heavy beast, and works best with some support or a gimbal. So the Sigma "Contemporary" for $989 is a much lighter and better choice for most. I don't know anything about the Tamron.
However... Everything can be bought "pre-owned" if your careful. And I always recommend the older Nikon 300mm f/4 AF-S (without VR) which can be found for around $550-$600. I still use that lens and it's one of my favorites. You just want to try and find a more recent serial number (the old ones had some compatibility issues with newer cameras).
My thought on losing money starting over is that it's like paying tuition. Just about everyone starts with DX because it's a cheap entry into photography. Then, if they like it and see other pictures it becomes "I want to do that!" So you start the upgrade path... and so it goes.
Oh, last thing. There is a guy you may know of, Steve Perry, he puts together lots of YouTube videos and has written a couple books on wildlife photography that are really helpful. He's also super helpful and you could email him and he'll respond to you. I really suggest watching his relevant videos here:
https://www.youtube.com/user/backcountrygallery/videos
And this is his website:
https://backcountrygallery.com/
So, that's my rambling for Saturday morning...
. I really want to upgrade. I hear the D7500 is pretty good, a lot better supposedly than the D7000. All my lenses are dx unfortunately as when I purchased them, I didn't know any better and wish I would have gone with the fx lenses as I am thinking I want to get away from crop sensor. I know I can use my lenses on the full frames. It is such an expense though to start over! I do like the lens other than for birds as it really is a great walk around lens.