Wow, you're a land lubber then, despite living in Valdez... :-)
OK, where to begin. Disregarding the drunk captain (who I'm told taught navigation in NY to mariners after the incident...), historically first and foremost you don't use celestial navigation if you have land marks to navigate by. That is to say, when in coastal waters, the sextant stays in it's coffer. When you're in coastal waters you use "piloting" skills, hence the name of the specialist you take aboard your ship for extra sensitive tasks. (I.e. the pilot). Piloting uses bearings to visible land marks (rocks, houses, light houses etc), depth readings etc. to establish your position, and compass and log to update it by dead reconing. In the age of technology many of those bearings could be taken by radio direction finding (of radio beacons), manually and via radar. The radar can of course also be used to get bearings and range to other (radar) visible landmarks. That's how ships could pilot in poor visibility, by using so called "radar navigation".
Now, you don't use a depth gauge (whether a knotted line, or a sonar one) to avoid a shoal. By the time you've detected it you're already either on it, or much too close for comfort. Instead navigation by depth reading is done to keep you i.e. in the middle of a sound, or comfortably far away from a coast etc. (Very useful when you have gently sloping costs, that are common in sandy regions. If you sail outside a beach, you can often rely on depth for navigation. Rocky coasts, not so much.)
Now, what to do when you can't navigate by landmarks. That's where celestial navigation comes in. If you have the proper knowledge (tables) a way to measure the apparent height of a celestial object (sun, moon, star) and an accurate clock (which ships have had since the late 1700-hundreds), then you can determine your position on the globe. Now, for reasons of simplicity most ships that still bother with celestial anything resort to a single noonday sun reading, as that's a simple thing to do, and doesn't require you to be as precise with your timing (and working out the position is simplified). Since you're in the middle of the ocean anyway, you don't need to a fix more often than that. You can safely use dead reckoning until next noon. However, should you want to (and have the correct tables) there's nothing stopping you from using the sun as long as it is visible at any given time of day (back in the day, sightings were often taken at least once a watch, i.e. every six or four hours), or the stars at any time they are visible. The moon is also usable, but since it moves so quickly over the sky in relation to the stars, taking a moon height requires skill and dedication, and hence navigational stars are preferred.
All this is historical of course. The merchant navy doesn't even carry the equipment to navigate celestially anymore, and they're not skilled in its use. Today it's GPS and everything else a distant second. Radar navigation skills are still practice however, as it can come in very handy.
So, when it comes to the Exxon Valdes, having your radar turned off if radar navigation thorough that sound was warranted, would indeed be a court martial offence. "Sonar" would have probably been of very little use, as a depth reading will only tell you that you're seriously off course (soundings on charts are only so accurate) and with a ship that size, you should already be well aware of that fact if you're to have any chance of saving the day. Even thinking about using celestial navigation in narrow coastal waters should have you forcibly removed from the bridge, as that couldn't tell you anything you needed to know in time to do anything worthwhile with a ship that size, moving that fast. Celestial navigation is for the open water, period.
P.S. We got the double hulls now and they're not all they're cranked up to be. You see, they don't actually use more steel in the construction of a double hull tanker. So it's the same amount of steel now divided into two thinner hulls, with an empty space you can't get to. Rust, however, can. So you now have a rusting time bomb that you can't inspect and that isn't that much sturdier than what you started with. Many believe the double hulls that are sailing since the Valdez spill are nothing less than ticking time bombs that will starts to go off any day now...