I'm wondering how to recover from losing my navy. Since ships spawn outside the port city, and there is no navy "city garrison" or militia equivalent, I'm having a hard time getting a navy back up. The enemy can just sit in sight range of my city and kill the newly built ships one at a time, because I can't dry dock them or protect them in some other way until I can get a reasonable navy going. Since it's unlikely that I'll have more than 1-2 port cities in the early-mid game, I can't even amass ships in another port.

Is there a way to deal with this kind of situation?

You could build city improvement "coastal battery" to do damage to close ships. Additionally I would buy some industrial boosters on marketplace, to fasten your ship production to prevent the production of one ship only in one or more turns. So you should have more than one ship available at once. If it works, depends on your enemy strength, waiting in front of your shipyard.