I've been doing rinseless washes for about 7 years, using both ONRs, Ultima, Meguiar's and Duragloss, constantly trying different media and technique.
Here's what I'm doing now:
If extremely dirty, such as salt-sprayed in the winter, I'll pre-soak with a pump sprayer. My media of choice is a Meguiar's or Blackfire MF mitt. I've tried the MF-covered Bone sponge, MF noodle mitts, grout sponge and MF towels. What I like about my current mitts is how easily they release dirt with just some hefty shaking in the wash bucket. I'll use them as a pad, not a mitt, doing 4 passes with the mitt, reversing direction on the same side while slightly lifting the dirty leading edge from the previous pass, then flipping the mitt for 2 more. Then the mitt gets a vigorous shake in the solution to dislodge any loose dirt.
Depending on the temp and sunlight, I may do a single panel or multiple panels before the next step. I'll follow up with a plush MF towel to blot large spots, spread out the solution and act as a backup in case I missed a spot. That towel gets constantly wrung out and reused, unless it did pick up any dirt. Next step is either a final pass with a drying towel or application of a drying aid, followed by wiping with a dry towel. I never use my dedicated drying towels when using any drying aids, as I don't want to compromise my drying towels with any chemicals or waxes.
With regard to your question of a dirty sponge, even with my technique, my white mitts will still be slightly greyed after the wash, since not every single microscopic piece of dirt is released. After a few washes, they'll never be pure white again, even after a cycle through the washer, but I've never found that to cause any swirls or marring.
Also, if I've only done one car with a bucket of solution and it wasn't too dirty, I may give the solution a day for the dirt to settle below the grit guard, then scoop out some clear solution from the top and fill my sprayer for the next pre-soak job.
There's a zillion ways to wash a car, but this seems to work well for me and two of my fleet are dark, one Black and one Dark Burgundy.
Bill