Yes, we did have a navy from 1775-83. It wasn't much. A handful of ships attempted to wage war against the British navy. Most of the "American navy" was made up of privateers. [Pirates to the enemy.] These ships engaged British vessels along the coast of America. [There was also Arnold's little makeshift navy on Lake Champlain.] Many of the privateers who were taken prisoner were sent to Forten Prison in Portsmouth, England to await prisoner exchange, parole or the end of the war.
Check state military records and histories to see what is available on these naval veterans. Also check for pensions.