Newly designed ships not saved unless game itself is saved
So I went and designed a brand new ship in the builder, but this time actually saved my game prior to exiting. And this time when I restarted the game and loaded the brand new save game, the ship was available.
I don't have to actually build the ship in my starport. I simply have to make sure I've saved the game I used to design the ship to ensure it's available in the future.
So note to everyone - make sure you save your game, not just your ship, after designing new ships.
-HM