My opinion, I think you'd be wasting effort in getting a system set up to get CO2 in the bottles before capping.
To answer your questions, I haven't heard 100% that the caps work, but I assume that they do, at least to a degree. The lining has a chemical in it that, once wet, absorbs the oxygen in the headspace. This doesn't happen immediately, so don't be concerned about getting them wet while dunking them in sanitizer and then becoming ineffective in a matter of moments.
As DrunkleJon pointed out, you'd have to flush the headspace in the bottle after pulling the bottling wand out, kind of a pain IMO.
With proper bottling technique, you can cap on very little oxygen, the caps giving you a little additional peace of mind. If you fill the bottle from the bottom with little aeration (as with a bottling wand), rest the sanitized cap on top, and let it sit for 30 seconds or so (I do this by filling a few at a time, then capping a few at a time), then crimp the caps, some of the CO2 left over from fermentation should theoretically push some of the O2 out. Again, this is very minimal at best.
I'd recommend using the O2 absorbing caps for beers that you plan to store for a while, use regular caps on everything else if you choose to save a little bit of money. You would not need to adjust priming sugar as was originally asked. The time and effort you'd spend on flushing the bottle headspace with CO2 isn't worth it, I very much doubt you'd see an improvement for that effort.
