No. Apparently "country" stems from "contra" as in "the land "opposite of ..." and refers slightly more to the land/territory while "state" stems form "status". So "state" is technically more a relation to the political entity governing a country, while country may refer more to the geographic territory. That being said they are usually used interchangeably as the state is tied to the territory and speaking of the interests and relations of a territory always refers to the political entity.
Though as you have or undoubtedly will realize there is no undisputed definition of what constitutes a state which has probably largely to do with the fact that this has certain ramifications with respect to the relation to other such entities.
Usually the problem is that each plot of land on earth has a flag sticking in it, meaning some entity lays claim to it, so that the creation of new states/countries can only happen through secession or revolution. Which then prompts the question as to whether the resulting structures from that are "legit" or not. Like does the sovereign state from which the new country seceded have the right to uphold the sovereignty within its borders. Like is it still their territory and is rebellion just treated as a criminal act or is the rebellion recognized as a de facto state that itself has the right to assert its sovereignty and should be supported in that goal. And there's a whole theory of states being states through being recognized by other states, though that entails the question what made these other states legitimate to begin with.
Or the other hand, is a permanence in territory and population entail that a state after a revolution tied to contracts agreed to before a revolution? So is it still the same state or a different state with many similar features?
Or yet another definition, can there be a country without a state? So if you put the focus on the "a 'state' is a polity that maintains a monopoly on the legitimate use of violence" or on it being centralized or whatnot, could there be collectives without a political entity representing them or that organize decentralized and thus "without a state" or is the state just the organization of people regardless of how that organization looks like?
So what is and isn't a state is a complex matter without a clear definition that settles that all, but no, it is usually not implied that there is a lack of sovereignty by calling it a state. It's the opposite of that.
By calling it a state you emphasize some sovereignty as "country" or rather a sub-country. So the difference is that a unitary state is one country and one central government for that country, so one state, while a federation is a whole bunch of countries with their own local governments and legislation forming a super-group that acts as one entity when interacting with other countries, yet where internally the constituents retain some sovereignty. And that relation can be tighter or loser, can be bottom up or top down. Or things like the EU or UN where the members are fully sovereign countries and where the Union, though technically binding to its members is not acting as a state or country of its own.
Also with respect to the USSR, that naming was likely just an attempt at positive branding. Like the initial idea of socialism as worker self-management and bottom-up organization. And also "soviet" just meant "council" and apparently used to refer to direct democratic workers' councils who organized the revolution. With the idea of a "soviet republic" not being too dissimilar to something like democratic federalism. So the "Union of Soviet Socialist Republics" implied lots of independent workers' movements rising up and joining forces to a powerful superstate, while retaining a bottom-up organization.
So calling subdivisions in the USSR "states" and referring to the country as a Union, rather than a country or empire, was probably meant to emphasize an independence and a mutual nature that likely wasn't there in the praxis.
So technically "states" does not imply a subdivision of a country unless you speak of a union or federation; on its own it rather refers to the independence of a domain. Though ultimately it depends on context and intention of the speaker and in different languages the connotations might vary.
For example, Germany and the U.S. are both federal states, but when people in the U.S. speak about the "federal level" they mean that something is centralized and decided on the level of the federation while when in Germany the "federal principle" is emphasized it's usually the opposite, meaning more privileges and responsibilities are given the the different states rather than the central government. So yeah likely no, but as always it depends.
