Plumper and Valiente make excellent points. In my opinion, the average sailor makes the same mistake that seamen have always made in fog; the desire to achieve the destination overcomes the practise of prudent seamanship. Particularly on a small sailboat (are there any other kind?), once in fog, the primary goal should be to avoid being run under or running aground. Often the best way to accomplish both is to make for skinny water. The other vessels that are going to kill you, and even if they're making bare steerage way they're going to kill you, will not be able to follow you into shallow water. And, even if you become disoriented, judicious use of the fathometer will ensure your safety. Many is the time when you do not have to know exactly where you are, just that you are out of reach of traffic and not aground. Motoring to safe waters is eminently sensible.
Val's suggestion reminds me of an out of style exercise that can be fun and rewarding. In good weather, have a below deck navigator attempt to steer you in a safe direction with reference to only a last know position, the chart, the compass, and the fathometer. You'll probably be pleasantly surprised at how well you can navigate. Generations of seamen didn't take all those soundings you see on the chart for nothing, eh?
“Scientists are people who build the Brooklyn Bridge and then buy it.”
Wm. F. Buckley, Jr.