I'd suggest you shoot with the new D500 for a while and not worry about "going full frame". I am certain you'll find the D500 a vast improvement over your old D200 (CMOS vs CCD and much more)!
To really get full advantage out of an FX D800 (36MP), you need to use it with FX lenses. Those are necessarily bigger, heavier and typically more expensive than DX lenses. While you can use DX lenses on the D800, doing so sort of negates the whole purpose of getting the FX camera. DX images from a D800 end up around 15MP... or less than 75% of the resolution of the D500.
The FX camera is ideal for wide angle work... while the DX camera is preferable for telephoto. Among the subjects you mention shootings, I imagine the D800 might be preferable for landscape, but the D500 might be better choice for the marching band. You mention "nature", but that can mean many things. If it's wildlife, the DX camera and telephoto lenses would be the better choice. If it's macro and close-up work, the FX camera might be a bit better choice. I use a full frame camera for some portraiture and studio work, too.
Plus, a lot depends upon what you do with your images. Unless you make big prints from your images you won't see much difference. You would probably need to print around 16x24" to start seeing the advantages of the FX format.
I suspect that a lot of FX fanboyz and fangirlz are actually the only people who see "all the goodness" from their larger format cameras. They see the advantage of FX while looking at their images "at 100%" on their computer monitors. Assuming their monitor is a typical modern type and is set to its native resolution, "100%" is actually a huge degree of magnification... viewing a 7,360 × 4,912 pixel image from a D800 at that is like looking at 6 foot by 4 foot print from about 18 or 20" away. By the time they resize the image for it's planned use, much of that "goodness" has disappeared! (Some of the above examples from D800/D810 are wonderful... but at Internet resolutions and sizes, 21MP images from D500 done with quality lenses would have looked just as good!)
The D800 is now an over 6 year old model. It has a native ISO range of 100-6400 (expandable to 50 and to 26500). It uses an anti-alias filter and can shoot at 4 frames per second. And it's 51-point AF system (15 cross type) is good to about -2EV and is "f/5.6 limited" (effecting the use of lenses with teleconverters).
The D500 is now an almost 3 year old model. It has a native ISO range of 100-51200 (expandable to 50 and to 1,640,000). It does not use an anti-alias filter and can shoot at 10 fps. It's 153-point AF system (99 cross type) is good to -4EV and is "f/8 capable" at 15 points.
Yes, the D800 has better high ISO performance . It also has more overall resolution (when used in FX mode, with FX lenses) and slightly greater dynamic range.
But in almost every other metric, the D500 is superior. You didn't mention low light shooting and will find the D500 to be MASSIVELY more high ISO capable than your D200! You also didn't indicate what lenses you have now... if many of them are DX that will influence your decision.
I assume you are looking at a used camera because the D800 is a discontinued model... It was replaced by the D810 (2014), which in turn was replaced more recently by the D850 (2017).
The D500 is a current model, so might be expected to be supported by Nikon for some years longer than a D800.
More comparisons:
https://cameradecision.com/compare/Nikon-D500-vs-Nikon-D800https://www.imaging-resource.com/cameras/nikon/d800/vs/nikon/d500/https://www.apotelyt.com/compare-camera/nikon-d500-vs-nikon-d800Ultimately, I'd suggest you spend more time with your D500 and learn to get the best out of it first, before adding another camera. I suspect you won't really need the FX camera... or the additional cost of FX lenses to use on it. But, who knows! Personally I use both full frame and APS-C cameras for different purposes. For what I shoot, I use the APS-C about 95% of the time.