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There was no 'Sea Lord' at this time - the term was not authorised until 1904. The Commissioners of the Admiralty (to give them their correct name) usually consisted of a mix of politicians and naval officers; at this time they were referred to as 'civil' lords and 'professional' or 'naval' lords. The senior Commissioner (a politician) was referred to as the First Lord of the Admiralty.
There were two First Lords during the time span you give - from 16 July 1788 until 18 December 1794 it was the Earl of Chatham. He was succeeded by Earl Spencer, 19 December 1794 to 18 February 1801.
The senior naval or professional commissioner (a post that later became known as the First Sea Lord) was Lord Hood who served July 1788 to March 1795. After Hood left the senior naval man was Sir Charles Middleton (May 1794 to Nov 1795 (he later returned as Lord Barham to be First Lord 1805-06) and then James Gambier 1795 to 1801.