Those three-piece rods ARE a POS, but that's because they are aluminum - they won't hurt your steel barrel - a good one-piece rod is preferable - I have two - both stainless. One I cut the handle off because I chuck it in my cordless drill to clean my barrel - the other I use for oiling and my rifles.
Fist thing, unload the gun, make sure it is unloaded, check again, then I remove the choke tubes and break it down to forearm, stock/receiver, and barrels. Then,I spray carb/brake cleaner down my bores (OFF the gun - don't let stuff like that contact the wood) and let it soak for a minute. Then I run the brush on the rod in my drill down and back, cleaning the barrels and the threads inside. Then I run patches until clean. I do the same procedure with my choke tubes - ( you can hold the tube in one hand and the drill in the other). You need to make sure you get any build-up out of your choke tubes. This buildup will be from the plastic wad and can cause a drastic change in choke constriction if you let it get too bad. Next, put some cleaner on a patch (I use Shooter's Choice, but many are good), and wipe down the receiver face and anywhere else where some gunk has gotten. Personally, I have a bunch of different dental picks to get into weird small spaces (there are nylon ones that work and no worry about scratching anything).
So far now I have cleaned the insides of my barrels and the receiver insides and my choke tubes. Now, I place some grease on the threads of my tubes and replace them into the barrel. Then I run a patch with a LIGHT oil coating inside the barrel. I place some oil on a patch and do the same to my receiver. I place a small amount of grease on the hinge pin, place the barrels back on the receiver, wipe down the outside with some oil on a patch, and put the forearm back on. If necessary, a small amount of wood furniture oil on a small cloth on the wood parts, snap caps go in, hammers tripped, back to the safe. Sounds long and complicated, but it really takes about 5-10 minutes tops, depending on how much cleaning the chokes need.