We are now contemplating what to do. If we bury the fence the full 12",
will that be enough? What other ideas do you all have, so we can do
something ONCE and have him contained and safe?

Consider adding chicken wire. Get 2" wide, and fold it down the
middle. Use clips (J-clips or hog rings work well) or zipties (the
black ones last longer but they all break down in the sun eventually)
to fasten the top 1' to the existing fence, and let the bottom 1'
extend out over the ground. Weight it down with a good sized rock,
brick, paver, etc. wherever it buckles, and wherever your dog has been
working at it. Let the grass grow up through it and it will anchor
itself down for you with a few weeks to a month. Or cover with mulch.

If there's only 1 or two spots he likes to dig, you might try just
throwing down pavers in those spots (in front of gates is a classic
spot).

And last but not least, get out there and play with him, so that it's
a fun place for him to be. Consider having him in the house and only
letting him out when he can be supervised. It does make your life
harder, but if the alternative is to spend the day hunting down your
dog after he gets out...

Sadly, none of these are a "guaranteed, will work every time"
solution, only a "this has worked before and may work for you, too"
sort of solution. For a sufficiently bored and motivated dog, there
is no fence that is a perfect obstacle.

Good luck,
--Glenn Lyford