Other than HFCS, refined sugar almost always from sugar beets. Not cane.
In the US this is true. But can sugar is a lot cheaper to import and grow elsewhere in the world. I think there's a few other sources of "table sugar" too, but beet sugar and cane sugar are the two main ones.
I think this is changing some though as the trade deals with Mexico is opening up import of cane sugar some if I remember right. Also the cut in corn subsidies probably has lessened HFCS price advantage in the US some.
But really the biggest factor in using "real sugar" lately is just the anti-HFCS sentiment. They're largely not doing jack shit to reduce the sugar they put into foods (and are actually resorting to worse means by trying to use ones that they can label as things that don't sound like sugar). A few companies have though (I recall General Mills was slowly reducing the amount of sugar in their cereals, they decided to do it drawn out as they worried people would notice the reduced sweetness if they did it all at once).
The reason that sugar, in general, is no longer used is that the US sugar industry's lobbying efforts have kept high tariffs on imported sugar, which artificially inflates domestic sugar prices. I imagine HFC might still be cheaper, even without the tariffs, but the difference would be significantly less.
Sugarcane is grown in the US, mostly in Florida and Louisiana, while sugar beets are grown all across the midwest and western US.
According to the USDA:
Unless all of that cane sugar is going into only baking cookies and Domino and C&H bags on grocery store shelves, I would imagine it would be used in soda production as well.
HFCS has other advantages in soft drinks, it mixes easier. That's actually a large part of why it was developed is to be more suitable for blending/mixing (soft drinks being a major one). The "high" mix (the 55/42 version) of HFCS is almost entirely used by the soft drink industry too (like 90% if I remember, but that was a few years ago so it might have changed since).
A lot of food now actually seems to use a variety of sugar. A lot of it is hidden (use names that don't sound like sugar). It's always funny to see "contains no HFCS" and then see it has like 5 different sugars.
I'm sure cane sugar is being used in some premium brand sodas and the Throwback versions, but even if it's almost half the total amount of sugar, it's probably not in half the sugar containing sodas - which themselves are a small portion of the total soda market. We need a lot more sugar cane than what the two or three states that grow it are providing.
Imports have opened up some, and I'd guess that's where a lot of the influx is coming from for the major brands. Produce it in Mexico and truck it to the US. Might be wrong though, and distribution is weird because a lot of them bottle it regionally. But since the cane sugar versions are still niche I could see them trucking them in from elsewhere that the production cost reduction would make up for the cost of importing it.