I am a veteran submariner from the US Navy - some of this will be conjecture based on anticipated technology, but the science should be fairly intact.
I do not anticipate that anyone will successfully design an atmospheric drone that can successfully, as a single unit, disable a modern destroyer. Understand that I mean a drone in the sense as they exist now - a remotely piloted fighter/bomber is potentially just as deadly as its manned counterparts. But the physical size of the aircraft is important, as it ultimately limits the size and muzzle velocity of projectiles - the Avenger gatling gun mounted on an A-10 can already stall the aircraft in flight. Hypervelocity means buggerall if the platform can't bring the weapon to bear and fire it successfully.
Lasers do change the game, but there is a power density issue. If you have a laser that you can mount to an aircraft and fire repeatedly, then the warship can easily mount one that is much larger and fires much faster. For instance, the Phalanx CIWS on US ships today makes the Avenger cannon look like a water pistol. Anything you can put in the air can be put in the water (in a much deadlier format) to destroy the thing in the air.
The real issue with drones is that we will not field them against an enemy who stands even a remote chance of being able to jam the telemetry. Drone strikes are a valuable tool that have changed air support forever, but radio is always radio, and sometimes you need someone to make a decision at the last possible second, or defy orders that should be defied. Command abort authority exists for a reason.
Submarines - while I understand them intimately - can not ever remove the need for a surface fleet. Active sonar will spot you every time - submarines just don't use it in combat against each other because it gives your exact location to the enemy submarine. But sonar buoys will always find it, and despite what you may have heard, they are slow. You can understand this almost instinctively - the surface ship can be bigger and have larger engines, but not have to face the sheer friction of solid water. Aircraft carriers can always outpace everything else in the fleet, simply because of engine size.
So the surface fleet will continue, in some form, until there is either no need for aggression, or we find a more effective heavy weapons platform.
I feel the need to clear up something about how target locking systems work, which works out to a game of planning ahead.
First of all, in order to achieve target lock, you need radar. When you point radar at a modern weapons system like a destroyer or a fighter jet, it immediately knows that it's in target lock and from what direction; it can also tell which kind of radar is pointed at it, and sometimes the exact model in use.
There's a sweep beam that finds targets, and a pencil beam that maintains a lock on the point of interest.
Now, on an Aegis equipped ship, the sweep is done with a phased array antenna that doesn't actually move. When active, it can and does sweep fast enough to respond to hypersonic missiles, and the Phalanx CIWS cannon can tear it apart reliably. This was one of the design considerations of the Aegis platform.
Now staying active has it's own risk, because you're lit up bright. Anti radar weapons just need to home in on the big radar source and don't have to send a pencil beam, but the Phalanx CIWS can home in on the radar return of a seagull. Not many problems here.
Now, having reviewed current and anticipated laser designs, if you put a thing in the air with a laser that can one shot a surface warship, it isn't going to be small. It's going to be phenomenally huge, and it's going to have - you guessed it - radar. As a general rule, radar on surface ships doesn't stay active, making them harder to spot.
So the exchange goes something like this. Drone is informed of a potential surface target; drone makes it's way to the target area; drone disappears because the US Navy was in touch with CENTCOM and satellite intel told them where to find the enormous flying laser, and the Navy laser has a longer range, higher power and faster repeat rate. (See what I said about planning ahead?)
This is the way war technology goes - a new tech becomes available, countermeasures and counter-countermeasures are engineered, and the game once again comes down to who can most effectively place their chess pieces.
"Drones" are great tools, but just one of many. For the foreseeable future, they are little balls of death you call on when an F22 squadron has better things to do, and the best payload I've seen so far is 3000 pounds. I'm not really worried yet.
Sailors have always been perfectly aware of the many horrible, agonizing ways they could die aboard their vessels. They have never been invulnerable, nor will they ever be, but we will always walk softly and carry a big stick.