Prop Dr.
An 18 wheeler size wrecker had problems pulling out an F-350 ? For Pete sake, I am thinking that "Riverside, CA" might be some kind of secret code for "A planet far far away with truck eating earth" I'm calling Tommy Lee Jones and Will Smith to investigate. Just watch out for the mind eraser bright light thing... (Of course this experienced may be something you want to have erased from your mind)
OK, on a serious note, It is plausible that the wrecker operator was not really up to speed but those trucks are easily over $200k. Normally the guys driving/operating them are the most experienced in the company, but I guess you never know. So, lets give him at least most of the benefit of the doubt and say that he was at least pretty decent on the controls and the crap your in really is that bad. - Given the experience you have had so far, I am thinking that might actually be a fair call/assessment.
If the wreckers main problem was that he had problems stopping his truck from being dragged toward the mess that is at least somewhat legitimate and may be helpfully diagnostic.
This tells us that any further serious winching or pulling efforts are going to be difficult at best. This tells us that we will really want/need to use that dead anchor with multiple anchor points plan (From the drawing I posted) if at all possible. If you can get at least 2 trucks out there (one with a winch) you can rig to use both of those truck as anchors. 3 trucks would be better if you have the snatch blocks and cable to use them... Of course your still going to want/need to proceed with the plan of digging out, filling in, mats to try to get back on the top of.. whatever that stuff is your in. If your running sort on cable length there is an easy way to make the most of what you have. I'll make up another drawing and post it for you, it might help.
You said you have made 4 "mats" if you can make 2 more (4 more is even better) you'll be well ahead of the game as you can then do like the ancient Egyptians (Except they used logs) and get up on a set, place a set in front of the ones your on, drive onto those, move the "old" ones to the front and just keep going until your 100% sure your on good ground. This is one of those cases where the metal sand mats would come in really handy.
I know there has been a lot of discussion about “Sand mats” or “Sand ladders”. I think this case is a perfect example of the justification/need for them. I found a place in California that makes them. However, as is consistent with your luck over the last few days, and California being a fairly large state, the place is like 450 miles away. The sections come in length up to 10 feet long. 4 of those would be ideal for getting you back to “Dry land”. Here is a link to the company, (Not like your going to run up there first ting in the morning, but thought you might find it useful for next time..
http://www.macraesbluebook.com/search/company.cfm?company=844302
Another company, probably the largest that sells them is:
http://www.calumetindustries.com/?cat=36 They have pictures too…
When your dug out and ready to drive (Move forward) if you can be attached to a winch or pull truck and move slow and steady onto your mats letting the pull bring you up on the mats as you gently assist with smooth application of minimum power, you do not want the wheels to spin if you can help it, (But you probably have already figured that out)
If you only have one anchor/winch truck if you can... And if the road has anything like ditches, move the winch/anchor truck to the other side of the road and into the ditch to help solidify the anchor point. Of course your going to have to close the road to do that. I have used this technique several times and have even on purpose "Stuck" the winch truck in the opposite side ditch. Note I said "Stuck" not "Buried" once the rescue is over and the truck is on good ground, you use that vehicle as an an anchor to pop the winch truck out.
(I assume you have already aired down the tires quite a bit??)
The lifting frame you designed was a good move. Very labor intensive but at this point, pretty much anything you do is going to be.
Air bags are an excellent idea. A great labor saving way to lift the truck. Having spent some time on the working end of a rescue truck I can tell you that the proper bags will easily lift a Deuce or 5 ton or even larger truck.
Renting bags will likely be difficult. I would be pretty surprised if your local rental center has them available. There are however 2 places that will have them. Your local Fire Department and the friendly guy with the 18 wheeler wrecker.
If you know somebody in the training division of a local fire department you MIGHT be able to get him to either bring some bags out or send some guys for a legitimate training session on proper bag use. Lacking that you might contact the wrecker company. They have bags they use for recovering overturned tractor trailers, they will be able to lift your truck while you back fill or place your mats. You'll probably have to pay by the hour but if you rig the winch truck and have the mats pre-positioned with some luck you might only need them a couple hours.
I hope some of this helps in some way....