This situation is a perfect example of when back-button AF (autofocus) is very useful. It is essentially why Nikon calls their AF back-button the 'AF-L' (Autofocus Lock) button on many of their more recent cameras.
When you have AF tied to the shutter button each time the shutter button is returned to the half press position, whether from the full press position or from the not pressed position, or each time a frame is exposed, the camera "resets" and will attempt to confirm focus has been achieved. In other words, it's going to refocus. If the scene has changed, then the focus distance will be changed. That's pretty much universal behavior with all SLRs/DSLRs I've ever used.
Short of using back-button focus...
You say that like it is a bad thing!
Back-button focus is a tool which many photographers find very helpful. It just takes a bit of using it to get used to remembering to always press the back button before releasing the shutter (fully pressing the shutter button all the way down) if the shutter button is set to not initiate AF.
But that is not the only way to use an AF back-button with most cameras that have options for back-button AF. It can usually also be set to lock AF when pressed and held.
Depending on the camera and the menu options, there's usually a way to use a back-button to lock, rather than initiate AF. In this scenario, a half press of the shutter button still initiates AF along with metering. When the "back button" (typically the AF-L button on Nikon cameras that have one, or the AE-lock button remapped to function as an AF-L button on cameras that do not have a dedicated AF-L button) is pressed, it holds AF at the current position until it is released. If the button is held over multiple frames, the focus distance will be the same for all of them.
This is the default setting for most Nikon cameras with an AF-L (autofocus lock) button. With Canon cameras, it is called an 'AF-ON' button because the default behavior is to initiate AF. Using menu options, the 'AF-ON' button on Canon cameras can also be set to be an 'AF-Off/AF-Lock' button and the 'AF-L' button on Nikon cameras can also be set to function as an 'AF-ON' button.
The Nikon D200 is an older model that is a bit different than more recent Nikon bodies that I've looked at. It has an 'AE-L/AF-L' (Auto Exposure and Autofocus Lock) button as well as a separate 'AF-On' button. If you want to use a back-button to lock AF that is initiated by a shutter button half-press, then you would need to press and hold the 'AE-L/AF-L' button on the back of the D200.
It's covered on pages 51-59 and 147-152 of the Nikon D200 User's Manual. The description of the 'AE-L/AF-L' button and custom setting for the function of the 'AE-L/AF-L' button are found on pages 56-57 and 156, respectively.
You might try changing custom setting 'a2:AF-S Mode Priority Selection' from the default 'focus priority' to the 'release priority' option. It might allow continuous drive mode without confirming AF between frames, but I wouldn't bet on it.
You could also try changing custom setting 'a5:Focus Tracking with Lock-ON' from the default 'Normal' to 'Long' to increase the amount of time before the camera refocuses if it senses the subject distance has changed drastically.
If you are using the MB-D200 battery grip with the D200, there's also a custom setting 'a10' that allows remapping the 'AF-On' button on the grip to function as the 'AE-L/AF-L' button. The position of the 'AE-L/AF-L' selector on the camera body would determine if it locks AE, AF, or both.
If the camera still insists on refocusing even when the 'AF-L' button is pressed and the camera senses that the subject distance has changed, your only option would be to select 'AF-On Only' for custom setting 'a6:AF Activation' and then press the 'AF-On' button to initiate AF. Release it once you've achieved AF on your intended target to lock AF over multiple frames.
I've never shot with a D200 or D300, but it would not surprised me at all that this might be the only way to maintain the same focus position between frames after using AF to initially focus the camera.
For a more complete list of different scenarios regarding how one may set up an AF "back-button" and in what situations each scenario might be useful, please see this answer to What does the AE/AF lock button do that half-pressing the shutter doesn't?