I have ships that have researched all items but can't convert ship XP to FREE XP. What constitutes fully researched?

Fully researched is when you have researched all items for a ship, so you have that right. To convert ship xp to free xp you will have to spend GP purchased in the market; it's not free to do.

Also note that a fully researched ship only need to have all of his modules and secondary weapons to be researched (unlock with xp) but not bought (with credits).