Good question, friend! Your gut feeling is right on the mark. The development of slavery in the European world, and its eventual shift into widespread use of African slaves, wasn't "sudden," nor was there a thousand-year gap between slavery in "Classical" societies and slavery in early modern ones. It was, rather, the result of a long historical process.

From what I have read, historians might assert that slavery had declined in some (some) regions of Europe over the course of the Middle Ages, but they would not likely contend there was some yawning abyss of discontinuity separating New World slavery from what had come before. To quote some excerpts from David Eltis' The Rise of African Slavery in the Americas:

"Both slavery and serfdom had been part of the social fabric of early medieval England, with just over 10 percent of the population classified as slaves in 1086. Even as those institutions died, new forms of compulsion evolved. The Statute of Laborers (1350-1) forced all persons not in a recognized occupation to serve in husbandry." However, Eltis also notes that "north and northwestern Europe had been free of full chattel slavery since the Middle Ages. Indeed the incidence of chattel slavery everywhere in western Europe had declined irregularly since Roman times, but the pace of the decline had been greater in northern than in southern Europe." (1)

So we see that there were definitely regional variations across Europe, and that slavery disappeared slowly rather than all at once. Eltis himself doesn't spend much time on this subject, since his book is more about post 1492 slavery than medieval slavery, but there are a few other books that might be able to provide the illumination you need. Allow me now to quote very extensively from David Brion Davis' seminal work, The Problem of Slavery in Western Culture:

"The problem of the decline of slavery in later Roman history is so entangled with the question of the decline of the Empire itself that one must be suspicious of any simple explanation. It appears that the condition of slaves improved at a time when the liberty of other groups was being eroded by economic hardship and the expanding power of emperors. While the servi benefited from legislation restraining their masters from the worst cruelties, the mass of agricultural workers suffered from oppressive fiscal policies and from the absorption of small farms into the great estates of the imperial aristocracy.

Yet the course of evolution from slaves to serfs and from serfs to free peasants was not as direct or invariable as has sometimes been thought. Slaves and coloni often worked together on a villa and were hardly distinguishable, but the wars of the fourth and fifth centuries brought fresh supplies of captives. Actual slaves still played and important part in the agriculture of Visigothic Spain, where the entire society was divided into two great classes of the free and unfree...It is nonetheless true that in most parts of Western Europe slavery declined and then virtually disappeared with the emergence of the feudal system. According to Marc Bloch, during the slow and profound transformation of the European economy, there was an overlapping and interpenetration of the systems of slavery and serfdom, so that servitude and dependencies of various kinds were modeled on the prototype of hereditary slavery, and such words as servus, Knecht, and vassus, which had originally implied unlimited bondage, acquired new connotations of status, rights, and willing service."(2)

So as we can see from Davis' account, slavery did not exactly disappear from Europe following the decline of the Roman Empire. I won't say it was just "given a new name," either, that would be simplistic. Rather, it seems slavery was subsumed--incorporated into a new social and economic order (feudalism) profoundly different from its predecessor, with chattel slavery leaving distinct imprints on the new order while having its own characteristics and even definition changed and reformulated. We thus come back to my assertion in the beginning of this essay: I think it's more accurate to speak of a gradual transformation that shaded into a decline of chattel slavery throughout north and western Europe rather than, as you rightly deny, OP, a straight-out disappearance.

Endnotes:

1: David Eltis, The Rise of African Slavery in the Americas (Cambridge University Press, 2000), pp. 4-10.

2: David Brion Davis, The Problem of Slavery in Western Culture (Oxford Paperbacks; Reprint edition, 1998), pp. 36-37.

I didn't directly cite them, but I highly recommend Davis' Inhuman Bondage, which is an excellent general history of slavery. He also cites some work by Youval Rotman, a French scholar of Byzantine slavery (Byzantium was the eastern Roman Empire, i.e Greece, what is now modern Turkey, etc). One of our dedicated historians of Byzantium would be able to tell you more about slavery in that region, but I'm sure they would recommend some of Rotman's work, like Byzantine Slavery in the Mediterranean World.