Shoot, more emphasis on an axle thru both. This will greatly help keep it all aligned. You may have to bang the backing plate in all the way and back out a few times progressively farther til it all comes apart.
I'd probably use gasoline and steel wool to get the gunk out. Final with brake cleaner and clean rags Lube the cam that expands the shoes well with grease. I woudn't work on the rust much -- the brakes might grab at first but the new shoes will clean it pretty fast. Where the shoes don't make contact it would be rusting anyway. If the broken off thing is outside of where the shoes make contact it shouldn't be a problem, otherwise it is
Replacing (or servicing) your drum brake (front or rear) is quite simple once you know the one important little trick involved. This is something you should do yourself as opposed to having a shop do it because you will do a better job. There are a couple other items you should address while in there. The pivot shaft that runs through the backing plate should be pulled, cleaned, and greased. Also, the brake drum surface should be cleaned. If you take it to a shop, they'll pop the old shoes off, stick the new one in, and probably won't do any of this other much needed stuff. OK, when you first pull the brake plate out, you'll probably be presented with something that looks like this .....
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Go to the auto parts store and get yourself a spray can of brake cleaner. You can do a rough cleaning first before you start the disassembly. If you're going to re-use the shoes, mark them "Top" and "Bottom" before removal so they go back in where they came from. Now for the little "trick" - the easiest way to remove/install brake shoes is to "fold" them off and on. Hold one shoe down in it's installed position and simply fold the other one up 90° to it. The shoes will easily come off now as a set still connected by the springs. Install is the reverse - set one shoe down in position (and hold it there), set the other at 90° (and connected with the springs) on the far side of the pivots, and simply fold it down into place. Once the shoes are removed, you can do a thorough cleaning of the backing plate, getting into all the nooks and crannies (Q-tips work well here).
Now on to the other things that need doing. The shoe pivot shaft should be cleaned and lubed. Before removing it, mark the split in the brake arm on the shaft head with a Sharpie. This will allow you to assemble it later with the arm properly positioned .....
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Once the arm is removed, you should find a felt ring seal nestled down into a depression around the top of the shaft .....
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Pull the shaft out of the hub noting the position and orientation of the thick washer just under the flats on the brake shoe end. Now you can carefully lift the felt washer out. Clean it by dipping in solvent and pressing between clean paper towels. Repeat until the solvent presses out clean. Knead the clean felt ring full of fresh grease before install. This is the seal for the pivot shaft and is what keeps the dirt and water out.
On the shaft, you'll notice it's center section, the part that resides in the brake plate, is depressed. This is a grease reservoir. After cleaning the shaft and hole in the plate, coat the shaft with fresh grease and work it in and out of the hole to displace the excess lube. Use a twisting motion so the depression remains full. Wipe off all the extra grease that wells up around the top of the hole.
Are you replacing the shoes because you've actually inspected them and found them worn too thin or simply because the brake no longer stops as well as it should? They may be fine and have plenty of life left in them, just needing a thorough cleaning and deglazing, along with the drum. That can be accomplished by lightly sanding the surfaces but by far, the best method is glass beading them. Here's some before and after shots of glass beaded shoes and drums .....
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The springs should be cleaned and if rusty, wire wheeled. Apply a light coat of oil afterwards, just enough so they shine. Before installing the shoes, apply a small amount of grease to the pivot post and the flats on the pivot shaft. After install, wipe away any excess grease that has squeezed out. The finished job should look something like this. Note the small amount of red grease showing at the edges of the flats on the pivot shaft (excess having been wiped away). And yes, this is the same brake plate assembly from the first pic after cleaning .....
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Edit 3/21/19 - Through careful research and measuring, member 2M has discovered that the brake cam is not exactly symmetrical. One end of the cam moves or applies it's brake shoe slightly less than the other end. The splined end of the cam has a dimple stamped in it. This denotes the end of the cam that applies it's shoe slightly less. The cam should be installed with the dimple on the inboard side, towards the axle .....
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All the details on this can be found here, starting with post #29 .....
http://www.xs650.com/threads/650-rear-drum-brake-plates.51046/page-2