The mic preamps are balanced differential: they take the *difference* between the signal on pin 2 and pin 3, and amplify that. Your mics put out a signal that goes negative on one pin, and positive on the other pin: identical, except that one is inverted with respect to the other. So the preamp takes those two signals, subtracts them, and ends up with 2x the signal on either one. X - (-X)= 2x. It operates on the _difference_ between the pin 2 and pin 3 signals: thus, "balanced differential".
On the other hand, EMI/RFI noise is common-mode: the same noise signal is applied to both pin 2 and pin 3. When the preamp sees that, you get Y - (Y) = 0, because it is common to both, not differential between them. The noise that is common to both pin 2 and pin 3 should perfectly cancel itself out, leaving *only* the differential signal that you want. So that's why reversing the polarity should not make any difference: when you swap pins 2 and 3 you should still get 0 for the common-mode noise, either way- unless there's a problems somewhere.
"Flipping the polarity" just reverses pins 2 and 3: it doesn't do anything at all to the common mode noise. In other words, it doesn't cause a normal channel to have +noise, and a flipped channel to have -noise, so that summing the channels will cancel the noise. You _can_ do that with in-band signals: audio _will_ cancel out, because the differential preamp can be reversed in polarity by swapping the input pins. But this noise is sneaking in through other means: perhaps coming in via the shields, and mixing in later in the preamp circuitry . By sticking a preamp with better common-mode rejection in in front of the board's preamps, you may be avoiding that problem.
The important thing is that flipping that polarity switch won't do a dadgum thing to the noise, since the mechanism that is used to flip it doesn't have any effect on noise that is sneaking in via the back door.
The old "pin 1 problem" comes when you have circuitry that has the ability to couple noise that is coming in on pin 1 (the shield ground) into the active electronics. This is really hard to design around in low-cost gear, since you have to make sure that signal ground and chassis ground are isolated from one another (or noise that rides in via ground currents can couple into your signal). Once again, it's common-mode noise- but it is getting in via a sneak path.
I wonder if you have some bad mic cables. I'd take the system down to its simplest state: one mic plugged in, ideally the one that contributes the most to the noise. Then, swap in the external preamp and see if that reduces the noise. Try a few different channels, too: there's nothing that guarantees that every channel will behave identically. You may have a cold solder joint inside the board on the input connector to one channel, or any of a number of other things. Swap the cable then, and see how that affects the problem. Carry the mic over to the board, and connect the mic case to the board's chassis with a piece of wire (essentially bypassing the shield in the mic cable). A few experoments like this should show you which link in the chain is the problem.
If you aren't right up against a high-strength RFI field, you shouldn't be having this problem: there must be an issue somewhere in the board's preamps. I don't know a thing about the Carvin preamps, but I can say that designing a mic pre for ideal common-mode rejection, and designing a pre for the lowest possible cost, usually don't go together very well...
