An adherent of Dispensationalism is called a "dispensationalist" but would the proper term for an adherent of Covenant Theology and New Covenant Theology be, respectively, "Covenant Theologian" and "New Covenant Theologian"? It sounds to me like it would lead to a little confusion, but I haven't really heard any terms in usage for either.
There are several options, but the terms "New Covenant Theologian" or "Covenant Theologian" are definitely not appropriate.
Those terms are reserved for Biblical Scholars, and is usually used in the context of pastors, or those who have advanced Theology degrees, such as a Master's of Divinity, or Doctorate in Theology, but it not used for a normal layperson.
I agree with the other person who answered that the closest would be "Reformed" or "Calvinist".
It has become more common practice for Calvinists to refer to themselves as "Reformed" even though they might even be a 5 point Calvinist or Hyper-Calvinist.
It's just that the word reformed doesn't have the "-ist" or "-ian" suffix that makes it a noun form, like Literalist, or Baptist, Dispensationalist, or Arminian.
Reformed or Calvinist.
The terms aren't identical with Covenant Theology, but they're pretty close to it. (There are varieties of Dispensationalism which are at least as diverse, if not more so, but they all bear a family resemblance)
In particular see the Westminster Confession of Faith, which includes a robust (and somewhat controversial) doctrine of the Covenant of Works:
The distance between God and the creature is so great, that although reasonable creatures do owe obedience unto Him as their Creator, yet they could never have any fruition of Him as their blessedness and reward, but by some voluntary condescension on God’s part, which He hath been pleased to express by way of covenant.
II. The first covenant made with man was a covenant of works, wherein life was promised to Adam; and in him to his posterity, upon condition of perfect and personal obedience.
III. Man, by his fall, having made himself uncapable of life by that covenant, the Lord was pleased to make a second, commonly called the covenant of grace; wherein He freely offereth unto sinnets life and salvation by Jesus Christ; requiring of them faith in Him, that they may be saved, and promising to give unto all those that are ordained unto eternal life His Holy Spirit, to make them willing, and able to believe.