The early copies of the LXX do is the divine name. Unfortunately we do not have copies of the New Testament that are as early as those LXX manuscripts. If we did then we would know for sure whether the early New Testament used the divine name. In the absence of NT copies the safest assumption is that the same practice of using the divine name was followed in the New Testament as in the LXX.
Those who insist that the NT manuscripts must be followed despite the fact that they date from a later period should bear in mind that those manuscripts don’t use “Lord” in full either, they all use abbreviations for Lord: KC. Those forms arose sometimes between the composition and the earliest manuscripts we have. So whatever way you look at it, none of event the earliest New Testament manuscripts preserve the original presentation of the divine name in those documents, they all contain a later modification.
This is the book where Lloyd Gaston agrees with George Howard that the New Testament contained the divine name and uses it in his translation of Paul’s letters.
https://www.amazon.co.uk/Paul-Torah-Lloyd-Gaston/dp/1597525383/