I keep forgetting about the challenge presented by a moving snake. I've occasionally wondered whether JETZEN shoots all those pictures of his snakes eating because swallowing a mouse slows them down enough for him to focus.
Your idea of setting the shutter speed at 1/125 and letting the camera go to the best aperture available is good. If that strategy doesn't give you at least an f5.6, you should consider increasing the film speed setting to get higher aperture and more depth of field.
Another trick you could try would be shooting with the flash but using something to diffuse the light from the flash. For instance, you could cover the flash with a layer of white plastic from a plastic grocery bag. The plastic bag may diffuse the light enough to give you softer shadows. You could experiment with other things like tissue paper or a translucent plastic from a food container. Another thought is to buy an actual photographer's light. They tend to be hot, and you'd have to have someone flick them on only long enough to take the shot. However, adding more light would allow you to get faster shutter speeds at higher f-stops and prevent blur from movement and blur from a small depth of field.
Ultimately, the most fun would involve using a camera with a flash sync and using off-camera flashes, but that's probably more trouble than you want to undertake.
Bill
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It's not how many snakes you have. It's how happy and healthy you can keep them.