Try Brian's experiment, too. You don't need to buy a separate body die. Just remove the bushing and expander both from your Redding FL S die and use its body only to set the shoulder back one or two thousandths at most, as Bart mentioned. That way you can still use the collet die on the neck.
If you don't have a proper gage for shoulder setback, you can improvise with your calipers and a spacer or a bearing bushing from the hardware store. You just want to measure before and after. The bushing doesn't have to match the shoulder datum diameter perfectly as this is a comparative measurement and not an absolute one. Just get something with a hole that lands about in the middle of the shoulder. A lot of 1/4" spacers are actually oversize to accommodate all 1/4" bolts and wind up being about right.
If you have a runout gage of some kind, check that your case heads are square to the rest of the case. It's not uncommon for a bolt face to be out of square or to be tipped slightly by bolt lugs not mating perfectly and in need of being lapped, or off due to an actual off-axis chamber. A head that's not chambering to the same fit each time can cause fliers by altering the rifle's recoil moments a little. I recommend you try to index all your cartridges with the headstamp upright when you chamber. That way if there is a head squareness issue, after the first time you fire them that way they will all be going back in with the head aligned the same way to reduce recoil moment variation.
Also check finished cartridge bullet runout if you can. Some guns care about bullet runout very little and others can open up an moa from it. If you have a way to measure it, you can drill a bullet size hole in your work bench and stick the bullet of the loaded cartridges in to gently straighten them. If it is not case heads but bullets causing an issue, it could be that 9 of your Collet die rounds seated straight but the other six were off slightly. Again, it depends how sensitive your chamber is to this.
I hope you are segregating your cases by which die did what to them so you don't cross over different amounts of work hardening of the necks by different sizing mechanisms. On another board a member recently posted some information from A-Square's book showing pressure and velocity increasing with each loading of some .30-06 brass due to work hardening of the neck increasing bullet pull, and with, start pressure. They even claimed up to a 10,000 psi pressure increase for their load. It seems high to me, but since they were measuring it obviously has the potential to happen.
That might explain why your FL loads had pressure signs the others did not. Your collet die should be working the brass less than the others as it neither over-resizes and expands, nor flows the brass to the rear of the neck. A standard FL die with expander should do the most work hardening of the neck. Given your pressure indications, it is possible this explains the results you saw. Generally, for cartridges firing at pressures high enough to stretch brass, it is the volume the case expands to during firing that affects peak pressure, not the resized volume. But the neck is another matter.
The solution A-Square promotes, and one a couple of makers of automatic annealing machines also promote, is to anneal necks after every firing. They are claiming to get the best velocity and pressure consistency that way. It's an awful lot of work to go to, but for the purposes of the numbers of cases you are testing, it would be another variable you could eliminate during load development. When you find the best load, you can try skipping it afterward to see if the groups open up or the pressure signs grow.
If you don't have annealing gear, I would use the candle flame method. It has a hard time overheating the brass. You just turn the neck and shoulder in a yellow candle flame until the head starts getting too hot to hold onto, at which point you drop the brass into water or wipe it off with a damp towel if you don't want to take time to dry them out afterward. The wiping gets the soot off.