Did Shakespeare really coin "Alligator"?

I have read many essays on the heavily debated subject of just how many words Our immortal Bard coined. I think it is safe to say, some of the words (and phrases) which are credited to him are undoubtedly his own, and yet there are as well many others which I, and many others, though wholeheartedly would love to see credited to Shakespeare, nevertheless, seem uneasy to accept it.

I was looking through numerous compiled lists of words supposedly coined by the Bard, and found one which truly peaked my interest–"Alligator"!

I already knew the etymology of the word due to my previous interest in the cónquistadors and Spain’s 15th, 16th-century history; initially "la lagarto" coined by the early Spaniards who entered The Americas and more specifically Terra de La Florida (possibly Ponce De Leon or those with him); and I already knew the Spanish word "aligarto" was already in use during Elizabethan England, and of course "Crocodile" is a word dating much farther back; But is there any evidence of the use of the word "alligator" before Shakespeare?

OED puts:

1560s, "large carnivorous reptile of the Americas," lagarto, aligarto, a corruption of Spanish el lagarto (de Indias) "the lizard (of the Indies)," from Latin lacertus (see lizard), with Spanish definite article el, from Latin ille (see le).
The modern form of the English word is attested from 1620s, with unetymological -r as in tater, feller, etc. (Alligarter was an early variant) and an overall Latin appearance

Here is the use of "alligator" by Shakespeare: From Romeo and Juliet, (V, 1, line 2840)

And in his needy shop a tortoise hung,
An alligator stuff’d, and other skins
Of ill-shaped fishes;

Here is the First (1623), Second (1632), Third (1665), and fourth (1685) Folio spelling:

and in his needie shop a tortoyrs hung, an Allegater stuft, and other skins of ill shap’d fishes

although Fourth Folio makes it "Allegator" with an "o".

And the first (bad) Quarto of R&J reads:

As I past by, whose needie shop is stufft With beggerly accounts of
emptie boxes: And in the same an Aligarta hangs, Olde endes of
packthred, and cakes of Roses,

But the second (good) Quarto of 1599 reads:

And in his needie shop a tortoyes hung,
An allegater stuft, and other skins
Of ill shapte fishes,

so, though the spelling is off, undoubtedly the word reads as "Alligator", and since those Jacobeans spelled the way a word sounded, I think there is strong evidence to hold that the pronunciation of the word is "alligator"–as the "r" in-between "aligarto" is moved to the front, "alligator" or "Allegater"–thus, the essence of our modern-day name of the animal is more like Shakespeare’s added "r" at the end" then any other archaic name used prior.

By 1643 the word "Alligator" is in use, as seen in Madagascar, the richest and most frvitfvll island in the world (1643)

the greatest enemy their Cattell have, is the crocodile or alligator, who being in the fresh rivers, meets with them sometimes when they come to water, but of these there are not many, and those might soone be destroyed if the natives had either wit or skill to doe it:

I did find the word being used in a 1542 text, but it does not seem to have anything to do with the animal:

or throwe any thynge agaynst the grounde or walles: [.] # alligator, he that byndeth: [.] # alligatura

My question is: did anyone use the name before Shakespeare in writing? And thus did Shakespeare really coin the modern name of "alligator" (even though it was spelled differently–different syllables, but otherwise very recognizable)?