Yes, you describe the initial condition of the saved.
But, this condition can be altered through e.g. habitual unrepentant sin.
Paul in Romans 6:16-19 gives a CHOICE to BACs:
Be a slave of sin >>> unto eternal death
Or ...
Be a slave of obedience >>> unto righteousness
And ...
Be a slave of righteousness >>> unto holiness
If you choose da blue, is this works-salvation?
No, I don't believe that it is.
It is not that the person is saved by any kind of work that they do; but it is that in being saved, their heart has been altered so that they have a specific kind of attitude towards sin and righteousness.
I'm not sure it pertains to the conversation; but I feel that it is noteworthy that Paul says that in context of what you have written above, he is speaking in human terms because of the weakness of our flesh.
The reality is that when we are born again, we are no longer slaves to sin, and that, it is weak terminology, thought acceptable, to say that we become slaves of righteousness/obedience.
But it is stronger and more accurate terminology to say that we have become sons of righteousness (Galatians 4:5-7).
A slave does what he does in the way of obedience because he has to.
A son does so because he is fighting for the honour of the family name and because he loves his family; his father, his mother, his siblings, and maybe even also his children.
For a son, love is the motivation for obedience.
For a slave, it is more often likened unto fear.
And fear is not an invalid motivation either. The fear of the LORD is wisdom (Job 28:28) and is the beginning of wisdom and knowledge (Proverbs 1:7, Psalms 111:10). See also 1 Peter 1:17 for a New Testament example.