Well, well, well, here's something I've got some experience dealing with! I spent the better part of 4 years in the Blue Ridge Mountains of North Carolina while earning a geology degree from Appalachian State University and followed that with 6 years of field work on mineral exploration and drilling crews, mostly in Louisiana, Mississippi, Michigan's Upper Peninsula, and the Piedmont of Virginia, NC, SC, and GA. In short, I've seen some mud, and I've spent beaucoups time digging trucks out of it. I've also dealt with a lot of sand during my dozen years of annual surf-fishing trips to the Outer Banks of NC.
Mark- our Ford Superduty OEM limited slip units are notoriously short-lived. I doubt yours is working. A rear diff service including the correct amount of LS additive might renew it but it might not. Mine's a goner, failing to work properly both before and after my last rear end fluid + additive change.
The "feathering the brake" trick works amazingly well. It's tough to finesse with a manual gearbox, however, so you run a chance of smoking your clutch. Ask me how I know that one.
You already carry one (or two) of the best possible "dead men", you spare tire(s). You're darn tootin' it's a lot of work to bury it.
Don't laugh, but a "Danforth style" boat anchor makes a GREAT deadman and it's much easier to excavate a hole for a Danforth anchor than for a spare tire. Might look a little funny Wandering the West with a boat anchor affixed to your brushguard, however. Get one suitable for a 24 to 27' boat and that will be fine for a fullsize truck, and there's even a hole in the shank for a D-ring.
A winch is super-handy once you have an attachment point, but the other 99.99999% of the time you're lugging around a lot of weight for no benefit. A good option is a high-capacity come-along with a snatch block. My come-along is rated for 4,000 lbs which I can double with the snatch block. The fact that a Superduty w/ a FWC is more like 9,000 lbs is not critical, as you're not trying to lift the truck off the ground, so even stuck, you're trying to accomplish much less of a lift than your truck's weight. Mine has 50' of 3/16" cable and I keep a chain and a couple of nylon snatch straps on board. Guys who hard-core the 4-wheeling and get stuck a lot need a winch. Guys who get stuck only ever 115,000 miles don't, but a come-along doesn't weigh a lot nor does it take up much space, and it comes in pretty handy around the ranch, no matter how modest said ranch is.
I don't think my high-speed scan of this thread caught any mention of airing down your tires--did you air down before or during the extrication process? Probably would have helped, and possibly a lot. I've given up on the "jumpstart battery w/ air compressors" after burning up 3 of them. I picked up a 100% duty cycle compressor from 4-wheel Parts and it re-pressurizes my 4 tires within minutes, making me entirely quick to air-down whenever the need arises.
Now for my own "getting stuck solo" story: I was driving the beach on Hatteras Island, the Outer Banks, in November 1990. By myself due to the sudden illness of my fishing buddy's Mom. Beginning or a nor'easter blowing, spitting a cold rain pushed by 15-20 knot winds. Low tide. Hadn't seen another beach buggy for an hour. The soft sand above the high tide line was a bitch to drive--coarse and heavily-rutted. The smooth sand below the high-tide line was SO inviting. Against my better judgement, down on the smooth hardpack I went. Everything was fine for a few minutes--I had the Trooper tires aired down to around 20 psi and that normally gave me very good grip. But the ground got softer and the truck started "cheating" downslope towards the Atlantic. No problem, just a quick downshift to 3rd (low range). Hmmm, still cheating, so quickly down to 2nd and a gentle left turn. Oh, crap, now she's REALLY cheating, and now I'm > halfway down the slope (still pointed perpendicular to the water, mind you, but the "cheating" was the downhill creep). So, threw her into Neutral and she stopped quickly, but without spinning her in. Got out, grabbed my short-handled spade shovel, measured the exact distance between the outside of my tires, and dug 2 shallow trenches in a gently curving upslope direction, all the way to above the high tide line, about 150', making sure to keep the trenches exactly the track width of the Trooper. Stomped up and down the trenches to pack down the coarse, pebbly sand as much as I could. Returned to the truck and dug forward from the rears to the front tires, packing that as best as I could. Aired all 4 tires down to 12 psi, as low as I dared, and still fearful I'd pop a bead. Unloaded a fair amount of my surf-fishing gear and walked it up the hill to above the tide line. Fired up the truck, let her warm up (all of this took over an hour), put her in 3rd gear (low range), and gently started letting the clutch out. She immediately moved forward and I was able to let the clutch out all the way within a couple of wheelbase lengths. Never spun a wheel all the way out. The waves were starting to wash over the tracks where the truck had been by the time I got turned around and finished re-loading the gear into the truck. She'd have been completely under water soon if not for the dig-out. The lesson I'd learned long before is that sometimes you have ONE CLEAR SHOT at self-rescue, so best to make it your BEST SHOT.
Glad to hear you're getting out and about even during these short days. I'm totally envious.
Foy