I'd also be happy enough to try other methods for controlling the crab
grass, but short of digging up my entire lawn, what options do I have?

If you've got real crabgrass, not different species:

You can get rid of your crabgrass with a bit of fertilizer, patience and some
better mowing techniques -- an old faq of mine:

Easy once you understand the biology of crabgrass, _Digitaria sanguinalis
or D. ischaemum_, annual species that germinate in bare soil in cool
temperatures, then doesn't grow much until the heat of summer comes. It
is intolerant of shade.

1) Fertilize and lime your lawn in spring and early fall, to help thicken
it. Best to get a soil analysis from your local master gardeners or
extension service, but you can probably wing it a bit with the help of
someone who knows the soils in your neighborhood well. If the grass forms
a nice, thick mat, it doesn't allow weed seeds to germinate. (Lime helps
correct the soil pH to 5.5 to 6.5, which allows the other grass plants to
take up nutrients easily.)

2) Obtain a wooden stake and put it in the ground someplace where you can
see it easily. If you've got the standard bluegrass/fescue lawn of most
of the northern states, make a mark at 3.5" and 4" from soil level. Set
your lawnmower blade to 2.75-3". When the grass hits the 3.5" stake mark
you should mow. At 4", you *must* mow. No cheating. If you remove more
than 1/4 to 1/3 the length of a grass plant in a single mowing, it weakens
the plant, possibly opening the soil for weed seed germination. Because
the grass stays comparatively long, it also shades the crabgrass seedlings
that have germinated and keeps them from developing.

Some times of the year, you'll mow a couple of times a week. Some times
(when it's hot), you won't mow for weeks.

The original research on this method of crabgrass control was done back in
the 40's or 50's -- the fertilizer used was chicken manure, and the
reported success was something like 70% reduction in crabgrass in the
first year, and 95% in three years. I tried it myself in a badly abused
lawn in a house I moved into... formerly a feedlot, more than 40 species
of weeds in the lawn. I used 10-10-10 fertilizer, the cheapest I could
find, a little glyphosate (roundup) on thistles and quackgrass (a
rhizomatous perennial) and some good quality bluegrass and fescue seed as
an overseeding in the fall.

I had virtually NO crabgrass in the lawn the following year, and, in
contrast to the neighbors who kept scalping their lawns "to avoid mowing",
I was down to 3 species of weeds within 5 years -- without major pain,
strain or suffering. The time I spent in the cooler months mowing was
certainly no greater than the time and effort the neighbors spent hauling
pesticides, spraying, cussing, digging, etc. And I was collecting
specimens for the weeds class I was helping with from their lawns, not
mine, because I had so few weeds, and so few species of weeds.

The lawn quickly reverted to a weedy mess after I moved out and the new
owners went back to scalping the lawn, opening up all sorts of bare soil
for weeds to flourish in.