Just to be clear, you cannot "go for an and one". And one means you're getting the continuation, which is where you are in the middle of a shot, fouled, but the ball went into the basked in that shot you were in the middle of. At least in theory, if you're changing your action after the foul, it's not a continuation so it isn't an "and one" situation. (In practice that may not always be the case, of course.)
So if a second player fouls you while you're doing that, odds are it's not a dead ball foul, but simply a foul, and you'll get one foul called (whichever they notice first or is worse) and one shot/pair of shots. That's assuming the second fouler initiated their play before the whistle (not making contact before the whistle, but starting the action that leads to the contact - jumping up, running full speed, whatever).
The player would have to initiate contact after the whistle (and probably a bit after the whistle, given reaction time) in order for it to count as a dead ball foul and get the technical free throws.