The other thing that makes a big difference is spraying under the bed, under the furniture and under all cushions on chairs and sofas, don't forget the cracks between the cushions as well.
I also sometimes cut flea collars into pieces and stuff them under cushions, inside of dog beds and other places fleas like to hide.
YES YES YES!!!!
You have already mentioned that you are buying all new beds and rugs, so that would be step number one.
Step number two would be to wash EVERYTHING you can in HOT water. Kill the buggers and start fresh. For things you can't wash in the washing machine (couches, mattresses, etc...) try to either have them professionally steam cleaned, or buy a steamer yourself (we have a Tobi) and blast them at high heat. For the clothes, linens and drapes that can be machine washed, pack them up in those nice space bags or vacuum bags and load them in the moving truck (obviously do the stuff you can live with out first and the things you need the night before the move last).
Jennifer is so right about spraying all the couches and mattresses (and do it as often as you can...) and putting cut up flea collars in cracks and crevices that can't be reached by kids and pets. If you can load them up before they can get re-infested you are off to a flea free new home.
Try to get the new home exterminated by a pest control company before you move anything in. As you said there is all hard flooring there, but it will help to insure that there are bugs hiding in the base boards or closets. Also, while they are there, have them spray the yards as well. You can spread the DM when you get there, but to start off fresh will be bestbet. I know you are trying to go as chemically free as possible, but to start with a fresh slate will make it easier to stay chemically minimalistic later.
For Jones, I would give her a nice bath and rinse at a do-it-your-self dog bathing place. While there, give a Capstar and then take her home to the new, flea free home. The Capstar will only work for the day, but it does work instantly and completely. It will keep anything from getting on after the bath and not allow any hangers on from hitching a ride to the new place.
The last thing I would think about is the car. If the dog is riding inside of your car, there are likely fleas there as well. I would have the interior washed and shampooed, seats if they are fabric and carpets, and them go over then with a steamer (like the Tobi) to kill anything in there. Toss a flea collar under the seats to keep them away as well.
FYI, we had a HORID flea infestation a few years back. I took in an infested foster and the infestation took off from there. None of my dogs had any major infection issues except Quinn, who gets a hot spot from one bite..., but all scratched to the point of hair loss. I ended up removing all carpets in the home and going all tile and wood. I did everything else listed above and at the end of the week, we were flea free. I continued to vacuum the furniture weekly (I hate dog hair) and re-sprayed the yards regularly, but other than that we just give the dogs their Sentinal pill monthly (it is also for heart worms) and the cat gets Program. We also had the new house treated as above when we moved a few months ago, and still have no flea issues.
Good luck, and hope you and Jones can have a fresh, flea free start in the new place.
Jessica