Romans 6 is talking about believers being dead to sin and alive to God. Your scripture has nothing to do with being a slave.
A slave is the same as a servant. There is a different kind of slave as one taken captive as spoils of war, but there is a contract of servitude hence slavery in the passive sense where by agreement between two parties, the servant serves as a slave for an advance for the family or means of provision for the family but Israel does have a 7 th year tradition where slaves or servants are set free from any binding agreement.
Now for the case in point about Romans 6th chapter; Paul points out whom you serve is the servant thereof and sin can be habitual in bringing a Christian into bondage again.
Galatians 4:8 Howbeit then, when ye knew not God, ye did service unto them which by nature are no gods.
9 But now, after that ye have known God, or rather are known of God, how turn ye again to the weak and beggarly elements, whereunto ye desire again to be in bondage?
Hence slaves to other gods in servitude and thus slavery to that sin.
Then there is Judaism.
Galatians 5:1Stand fast therefore in the liberty wherewith Christ hath made us free, and be not entangled again with the yoke of bondage.
2 Behold, I Paul say unto you, that if ye be circumcised, Christ shall profit you nothing.
3 For I testify again to every man that is circumcised, that he is a debtor to do the whole law.
To be a debtor to the whole law is to be a slave to the whole law and thus in bondage.
2 Peter 2:18 For when they speak great swelling words of vanity, they allure through the lusts of the flesh, through much wantonness, those that were clean escaped from them who live in error.
19 While they promise them liberty, they themselves are the servants of corruption: for of whom a man is overcome, of the same is he brought in bondage.
20 For if after they have escaped the pollutions of the world through the knowledge of the Lord and Saviour Jesus Christ, they are again entangled therein, and overcome, the latter end is worse with them than the beginning.
A Christian can err by sowing to the works of the flesh in reaping corruption and be overcome that they wind up worse than they were before by being a slave to sin once again mayhap more so where they do not care to ask Jesus to be delivered from their sins and to keep them from their sins. Any carnal elievers out there that read this, if you are pricked in your heart, you better call on Jesus now to help you repent because putting it off tomorrow, one may find the day where you do not care to repent at all and that sin rules over your life as you reap corruption and be worse than you ever was before. If you reached that point but still able to call on Jesus to save you from your sins, then do so now.
Okay back to our regularly scheduled programming....I do digress....
Anyway.... whom you serve is your master and thus servant or slave.. it applies.
Please do read my scriptures. If I am a servant to you then I am a servant to you because I have chosen to be one, not made to be one. I am not a robot and if Jesus did not give me a mind of my own and free will of my own, I sure would never serve a tyrant!! My earthly dad tried that with me, and that is why I was such a bad kid! LOVE has an open hand never a closed fist. We disagree much as to the true meaning of scripture it seems. Oh well, if you are going to correct me use ample scripture, we do not twist it to fit our morality.
One question for you to answer; how do you apply this saying from Jesus?
John 15:13 Greater love hath no man than this, that a man lay down his life for his friends.
14 Ye are my friends, if ye do whatsoever I command you.
15 Henceforth I call you not servants; for the servant knoweth not what his lord doeth: but I have called you friends; for all things that I have heard of my Father I have made known unto you.
16 Ye have not chosen me, but I have chosen you, and ordained you, that ye should go and bring forth fruit, and that your fruit should remain: that whatsoever ye shall ask of the Father in my name, he may give it you.
As it is, I hope in Jesus that He has chosen me since religiously keeping my former commitment to make Him Lord of my life did not work, but trusting Him as my Good Shepherd to help me to follow Him ... does work for He does work.
If we had to surrender from al our efforts in trying to save ourselves and believe Him that by believing in Him we are saved, then it stands to reasons and faith that we have to surrender from our efforts & iwll power in following Him to believe Him to be our Good Shepherd to help us to follow Him because that is what He said He will do per the New Covenant.
Trust is the basis for all relationships, and Jesus is asking us deny ourselves as able to follow Him as the same for saving ourselves and to trust Him to help us to follow Him while resting in Him that we are saved since we had believed in Him.