The entry for enfetter in Samuel Johnson's A Dictionary of the English Language (1755) cites Shakespeare's use of the word in Othello:
To ENFETTER v. a. {from fetter.} To bind in fetters ; to enchain. [Cited example:] His soul is so enfetter'd to her love./ That she may make, unmake, do what she list. Sh. Othello.
But the term also appears in a number of books in the period immediately after Shakespeare but before 1650. Here's a brief survey of "shortly after Shakespeare" usage of enfetter and its variants, as culled from searches of Google Books and the Early English Books Online database.
From John Speed, The History of Great Britain under the Conquests of ye Romans, Saxons, Danes and Normans (1614):
If Pope Paschall (in the time King Johns grandfather) hauing with much solemnity made some graunts to the Emperour Henry, and confirmed them with an Anathema, with the oathes of thirteene Cardinals, and with religious receyuing of the blessed Sacrament ; yet, because such grants were thought preiudiciall to his See, solemnly disclaimed his owne Act, and such his doing was aproued by a Clergy Councel, as pretended to be done by feare : how much more iustly might King Iohns Successors and his State, by such approbation of their grand Councel, free themselues of those seruitudes wherewith by anothers vniust, forced, vnwarrantable Act, they were supposed to be enfettered?
From George Sandys's traslation of Ovid, Ovid's Metamorphosis Englished, Mythologiz'd, and Represented in Figures (1632):
He's mine! the Nymph exclaim'd : who all vnstript;/ And, as she spake, into the water skipt:/ Hanging about the neck that did resist;/ And, with a mastring force, th' vnwilling kist:/ Now, puts her hand beneath his scornful brest;/ Now every way invading the distrest:/ And wraps-about the subject of her lust,/ Much like a Serpent by an Eagle truss't;/ Which to his head and feet, infettered, clings;/ And wreaths her tayle about his stretcht-out wings.
From Robert Bolton, The Saints Soule-Exalting Humiliation; or Soule-Fatting Fasting (1634):
But the master that thou servest, the Prince of hell, feeds thy soule continually with ranke poison, scourges it with fiery, invenomed Scorpions, (though for a while thy feared and senslesse conscience feele it not) enfetters it in the invisible chaines of darknesse and damnation : and after a while eithout timely repentence, and returne, will locke it up for ever in the dungeon of brimstone and fire.
From John Grant, Gods Deliverance of Man by prayer. And mans thankefulnesse to God in prayses. (1642):
When his wrath is kindled but a little, O blessed are they that in him repose their trust, Psal. 2. 12. accursednesse shall enfetter all others in the chaines of everlasting darknesse. O! my beloved in the Lord; What ever we have been heretofore, be this hence forward to the end of our pilgrimage, and warfare on earth, our mainest care never any more to receave the grace of God in vaine, never turn the same into lasciviousnesse, which Saint Iude fitly calls, the denying of the only Lord God, and our Lord Iesus Christ: ...
From Simonds D'Ewes, The Primitive Practice for Preserving Truth (1645):
All which open affronts, the Popes in this fifteenth age after our blessed Saviours incarnation, endured from these Kings; not because they were more deare to their Subjects then their Predecessors, or the Popes lesse potent then in former times, (for their strength in Italy was more encreased in that age, then in ten fore-going) but indeed it was the light of the Gospel that began about these times to dawn every where, that made way for dispelling those chains of darknesse, with which both Prince and people had in those former ages been enfettered; so as the Pope fearing, lest all should fall from him, as some Germane Princes, Republiques, and Cities had already done, was fain to comply with the French King, to submit to the Emperor, and to court the King of England, by the intercession of foraine Princes for a reconcilement.
As these five results demonstrate enfetter appears in at least five unrelated different books published from 1614 through 1645. Slightly later writers used it as well. For example, from Charles Molloy, De Jure Maritimo Et Navali ; Or, A Treatise of Affairs Maritime and of Commerce (1676/1682 [third edition]):
But in regard Masters might not be tempted to engage the Owners, or infetter them with such sort of obligations, but where there is very apparent cause and necessity, they seldom suffer any to go Skipper or Master but he that hath a share or part in her ; for that if monies or provisions be taken up, he must bear his equal share and proportion with the rest.
So while enfetter may be considerably less common than fetter as a verb and may owe its earliest use to considerations of meter, it is hardly a one-and-done production of Shakespeare's brain.