Actually, I think that chunks of solid food in soup which were intentionally put there by the chef would qualify as lagan, not flotsam. I know of no reason to think that Miss Manners objects to soup lagan.
Again, dumplings fall into the category of stuff intentionally put into the soup by the chef, and therefore may be eaten (if they’re edible). The other stuff, though, nope. (Remember the classic comic semantic mangling of this etiquette rule that went “Never break your bread or roll in your soup”?)
AFAICT, etiquette frowns upon the diner’s putting anything in his/her soup except salt, pepper and the spoon (and possibly some designated garnishes if they are provided—and it’s only fair to note that some etiquette writers specifically include small whole oyster crackers in this category). Like sopping up gravy or sauce from your plate with bread, crumbling or dunking baked goods in your soup just strays too close to the line of “playing with your food”.
Mind you, I loves me some soggy bread in soup or sopped-up sauce on a bread slice when I’m eating informally, and I don’t feel personally affronted in any way if other people like to enjoy them that way in restaurants too.
But the OP did ask if there was “a `right’ method” to eat crackers with soup. So I think it’s relevant to point out that according to traditional table-manners etiquette, the answer to that question is “Yes: separately”.
ETA: Oh by the way, we were talking about this just a couple of years ago.
I don’t know if it’s exotic exactly, but I crush them while they’re still in the little plastic sleeve, and then open it and dump the crumbs into the soup, that way I don’t have cracker crumbs all over me.
I absolutely love putting crackers in my soup. The crunch is a great addition to the whole soup experience. I find myself really craving the crunch so sometimes, if I cant find crackers, I’ll even put Goldfish! Has anyone tried that before? It works because of all the different kinds of goldfish there are too.