I know that the shape of the hull is too narrow for a medieval ship and I'm not sure if I should continue and make more of those or try a different approach…
Not so, the cog was a somewhat outdated design by the 15th Century, but still used. Other vessels were being introduced, like the crayer and hoy, which were somewhat slimmer than the traditional cog. Caravels and Carracks were becoming common too. As long as it's ship-shape (see what I did there?), it will fit as something.
Fore and aft castles were sometimes only temporary structures on some types too and only added if any fighting was likely.
So to me it looks the part at least. Nice job!
