Did English/British merchant sailors receive written testimonials between 1500 and 1750?

I'm going to go with a "no". Having investigated some different research lines here, nothing gave a conclusive "yes", so I think this wasn't a systematic practice even if some captains may have engaged in the practice.

Reasons

• The history of performance reviews was generally inconclusive;
• The history of performance reviews in the Royal Navy was inconclusive;
• Captain's frequently kept sailors in wage-servitude with the sailors spending more money on a voyage than they made (more below);
• The duties of captains and other officers (though in the Royal Navy; perhaps merchant behaviour was different) did not point to such habits (more below);
• Overviews of sailors' recruitment levels did not show problems with shortfall (more below);
• Overviews of sailors' recruitment methods did not indicate a "return to service" practice that would be enhanced by an 'official' performance review/testimonial (more below).

I accept that none of these are conclusive, and, as mentioned above, note that a practice of issuing testimonials may still have existed.

Officers' Duties (RN)

I did not see a comparably comprehensive overview for the merchant ships (especially hoping that the HEIC vessels might have had a method for this), hence this is based on the Royal Navy. The below noted a captain's duties for a mid-18th century frigate.

The captain was in overall command of his vessel and its crew and was responsible for its sailing, manning, and upkeep. Before sailing, he was expected to oversee the assignment of ratings to the members of the crew and to draw up and post ‘watch,’ ‘division,’ ‘station,’ and ‘quarter’ lists. He was expected to obtain from the Clerk of the Survey a book listing the inventory of stores allotted to the boatswain, carpenter, gunner and purser of his ship and to confirm that it was in agreement with the individual inventories of those men. He was not permitted to make alterations to the spars, sails, or hull of his ship. Finally, he was expected to keep a complete journal recording the activities of the ship and its crew and to sign and submit a copy to the Admiralty and Navy Office after each voyage.
_{—Flynn, 'H.M.S. PALLAS: HISTORICAL RECONSTRUCTION OF AN 18th-CENTURY ROYAL NAVY FRIGATE'}

Alternatively, this could have been the purview of the master, but his duties are listed as:

The function of the master was to assist the captain in overseeing the fitting out of the ship. He was expected to oversee the loading of all stores, and to report any damaged goods to the captain. He was in charge of the receiving, loading, and distribution of ballast; he supervised the loading of the hold, and continually oversaw the redistribution of stores over the course of the voyage to ensure the ship’s trim. He was charged with ensuring that compasses, glasses, log, and lead lines were kept in good order, and was responsible for navigating the ship in accordance with the orders of his captain or other superiors. He was further charged with observing all coasts and waterways and recording any new navigational details observed. When at anchor, he was responsible for keeping the hawse clear of fouls and obstructions. Finally, the master was expected to monitor and sign the accounts and logs of those below him and to ensure that he was thoroughly acquainted with their contents. Like the other officers, the master was required to supply himself with the necessary maps, instruments, and books of navigation and to keep a journal to be turned over to the Admiralty at the end of each voyage.
_{—Flynn, 'H.M.S. PALLAS: HISTORICAL RECONSTRUCTION OF AN 18th-CENTURY ROYAL NAVY FRIGATE'}

Nothing in the above describes anything related to sailors' performance reviews or assessments of any other kind. Admittedly, naval sailors would not have been released between journeys to go home except when they had leave, so perhaps this is not a very good comparison. Nevertheless, it sounds like a policy the Admiralty would have liked to enforce (jokes about military bureaucracy allowed?).

Sailor Recruitment Methods

Sailors were typically recruited rather than born into their communities and the various methods of recruitment for manning sea-going vessels affected the resulting demographics of the community. ...

The ideal method to cover the manning requirements of a vessel was by voluntary recruits, and this method was most successful for enlisting commissioned officers during the Anglo-Dutch wars of the seventeenth century. ... In contrast, efforts to encourage volunteers for lower-ranked positions in the fleet was often less productive. The men needed for these positions would not enjoy the financial rewards and status associated with the ranks reserved for “gentlemen”, and their work was often hard and considered menial. Yet, popular broadsheet ballads commonly pandered to the working classes in order to motivate voluntary recruitment.

Many working class sailors enlisted to escape poverty rather than to earn money. ...

The need for bed and board may explain why some volunteers came directly from other ships without staying in port...

The need for bed and board may explain why some volunteers came directly from other ships without staying in port... Indeed, poverty was likely the motivating factor for the majority of lower-ranked men on ships in addition to those workers whose voices are not recognized in official documentation such as female servants, child workers and indentured peoples.
_{—Delgado, 'Ship English'}

This suggests (but of course is not conclusive) that there was no lack of sailing work; that merchant sailors who found in need may have joined the RN; and that in general, sailors who were looking for work should have been in a position to find it. Also, their previous experience would have been apparent as soon as the vessel sailed so they wouldn't have been in a position to cheat by yet.

At the same time, if the sailors were in constant need of money (the references above also suggest that most captains used trickery to keep the sailors owing money, hence perhaps treating with those sailors on an even basis in the future wasn't a big concern:

The various ways in which the seaman's body was made to absorb the uncertainties of sea passage ultimately served as insurance to guarantee the merchant's accumulation of wealth.

Captains cut further into mariner's ages by selling basic necessities to seamen, often at enormous profits. They sold brandy, rum, wine, additional food, sugar, tobacco, caps, coats, shirts, trousers, breeches, stockings, shoes, and thread. They made further deduction for the Greenwich Hospital fund and the surgeon's fees. Master John Murrin asserted that "its usual for Masters of Ships to make profit of[f] what goods they sell to their Marriners." The costs of such goods were always deducted from the tar's wages, and not infrequently did a man "make a Bristol voyage of it," using up more value in these necessities than was due him in wages at the voyage's end. Such arrangements gave the master a claim on that seaman's labor for the future.
_{—Rediker, 'Between the Devil and the Deep Blue Sea: merchant seamen, pirates and the Anglo-American maritime world, 1700-1750'}

The above is perhaps the strongest evidence I have in favour of this supposition that such letters would not have been needed.

Demand for English Sailors

English sailors also seem to have been generally considered skilled and most foreign nations would have been happy to employ them (though the English discouraged this practice):

A crew which was largely made up of Englishmen might also mean that a ship could pass itself off as English, at a time when the Royal Standard was widely respected in the Mediterranean. ... the governor of Zante was astonished not only by the agility of English ships, but also by the ability of their crews who were accustomed to sailing even in the middle of winter.

... the men who worked in the shipyards were regarded by contemporaries as highly skilled and specialised. English sailors were thus much sought after throughout the seventeenth century, brought on board the ships of other nations by force, through deception, or lured with the promise of better pay.
_{—Pagano De Divitiis, 'English Merchants in Seventeenth-Century Italy'}

Hence, it looks as if employment wasn't a problem (though perhaps 'good employment' was...).