I think there are two issues here. One is what we do once we've poured out a bottle of beer. I triple-rinse that bottle, a little water, swish and pour, swish and pour, swish and pour. Three times works for me.
The second is how to clean dirty bottles. Mostly that's been covered, soak in a good cleaner, usually does the trick. If you rinse them well prior, as above or better, there shouldn't be much if anything to have to clean out.
In OP's case, sounds like the carb drop is at fault, as if it didn't completely dissolve. He figured that one out.
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I put the rinsed bottles in the (gasp!) dishwasher, neck down in the top rack, angled so the sprayer can hit them. They come out sparkling. Whatever sheeting agent is used doesn't seem to affect the head of beer in those bottles, but then I also rinse them with Star-San using a vinator just prior to bottling. This way they are cleaned as I use them.
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Related subject: had my first infection in beer I bottled back in October. A porter w/ a fair amount of nonfermentables. Chilled and opened one in late march, gusher! Opened a couple more, gushers!
Don't know why, though I had trouble w/ foaming in that beer before I kegged it. The OG of that batch was 1.020, but the gravity of the foaming beer was 1.010. Something ate through some of the nonfermentables in that beer. Grrrrrr.......
My son, in graduate school in microbiology (not studying yeast, sadly), said to throw those bottles out. He said there likely was a biofilm in them and that biofilms are notoriously difficult to clean.
Something about if they had such a thing on his lab's glassware, they'd probably just toss it rather than risk using it.
I tossed those 12 bottles into recycling.