I'm not aware of a fully socialist country, much less a fully socialist democracy. No, really, look it up. The wikipedia still has the clear, original definition of "socialism": the government owns every store, factory, and farm. Every business and employment of every kind is a government department. That's what Russia was "soviet socialist" republic. "Communism" is about living communally, eating communally, like the Spartans and the Amish.
So Americans looking to denigrate promoted both words downward. Socialist Russia (which never reached 100% socialism, there were always private businesses) was called "Communist Russia", though only a few of their farm communities were communes. And countries with high levels of social services like Sweden got tagged with the "socialist" label, when it was a wild exaggeration for having some government departments much larger than American counterparts...rather the way America had a much larger department of defense than Sweden. (I consider it harmless, fun,trolling to refer to America's "Socialist Army" and "Socialist NSA", forgive me.)
Every time I hear countries with higher levels of social services than America called "Socialist", I check Wikipedia again to see if the dictionaries have thrown up their arms and accepted this as the new meaning of "Socialist", but, nope, you're still saying that Canada must have a Government Department of Food Growing and the Government Department of Car Making.
Now that America is waaay out there as a bizarre outlier in medical care systems, honestly, it's America that should have the funny adjective in front of the name. But we need to ditch the "ist" adjectives that are just too sweeping and prone to wrong interpretation. "Low-Service Democracy" vs "High Service Democracy" ?
OH, and incidentally, if you imagine that your government doesn't "Control the Health Care System" in America, you're blind. Medicare may only be for 65+, but because those are the people that need the most care, that's half the medical work. Not to forget that the government licenses all the physicians, approves all the drugs and medical equipment, and on and on. They control it without paying for it. They call the tune without paying the piper...generally a bad feedback loop.