Can "still" and "yet" be used interchangeably when placed at beginning of a sentence?

Example:

The three picked up their chopsticks and started on the food. The scene seemed strangely normal to Naomi. It was as if everything were OK, and her mom were perfectly healthy. In fact, she looked more lively and energetic than Naomi remembered. The whole illness issue seemed like a distant, bad dream. Something that belonged to another reality.

Still, Naomi couldn't take the thought out of her head, so she decided to ask.

If I changed still for yet, would the sentence mean the same thing?

No. What first comes to my mind is that, even though they are very similar, yet connotes contradiction, whereas still carries the notion of defiance.

Let me view it from a different aspect.

Yet entails an element of surprise by the contrasting circumstances. Yet is like saying to your chum: "So, yeah, that's all nice and good... but check this out."

Still is more in the vein of: "So, yeah, that's that... but never you mind that."

It's a tiny difference, but it's a difference in the level of deductive capabilities that you, as the author, expect your audience to have. If they sense you consider them not capable enough of figuring out things without your solicitious guidance, and that you therefore opt for "yet" (... —Oh, yeah? What could it be?) instead of "still" (... —Yeah, yeah, go on, I follow you.), they might turn their backs on you. And stop reading your story.

In this context, still means ‘nevertheless, all the same’. Yet can have a similar meaning, but not when used in this way. Still can stand on its own at the beginning of a sentence and affect its interpretation, but yet cannot.