The Greek seems abundantly clear to me :
... but ye obeyed out of (ek) the heart unto (eis) that ye were delivered - a form of doctrine.
ητε δουλοι της αμαρτιας υπηκουσατε δε εκ καρδιας εις ον παρεδοθητε τυπον διδαχης
Romans 6:17 [TR undisputed]
The logic of the sentence is 'out of the heart' 'unto the doctrine'. The matter of 'delivery' is how the doctrine was administered to them.
I cannot see how anyone could misunderstand this but some translators seem to have done so.
παρεδοθητε is second person, plural, aorist 1, indicative passive (1). 'You were delivered'.
But the subject of 'you were delivered' comes after the verb : 'the form of doctrine'.
I think it is the Greek word order that has tripped up some English translators. But I see the KJV to be correct - in logic and in translation.
But, at the end of the day, it does not make a difference big enough to argue over. The doctrine of Christ is delivered to the church. And the doctrine of Christ delivers the church.
However, there is a difference in emphasis. 'The form of doctrine delivered you' agrees with the sentiment expressed by Jude -
... that ye should earnestly contend for the faith which was once delivered unto the saints
Jude 3 [KJV]
If anyone can prove the grammar to be otherwise, that is to say that in Romans 6:17 it truly should read 'the form of doctrine into which ye have been delivered' (D-R), which sounds exceeding strange to my own ears, then, if the KJV translators are truly incorrect, it may have been Jude 3 which they had in the back of their mind.
(1) Bagster's Analytical Greek Lexicon p 307.