I FINALLY FIGURED IT OUT! I'm going to go ahead and write about it here, should anyone else in the same boat find this page. I know I was searching all over the internet for answers.
First of all, my radio must be different than most of the other Honda radios out there. I kept reading accounts of "ERR" messages, which I never did see. I would occasionally get a random beeping sound, but otherwise nothing ever changed-- nothing more than the word "code" and a flashing light. I couldn't turn it off, as I said before, which I also kept reading about people doing. There is a lot of helpful information
here and
here. Many people mention turning the radio OFF and pressing 1&6 together while powering it on to get the serial number for the radio. Now, I didn't need the serial number anyway, but I was perplexed, because I couldn't turn mine off at all. The guys that fixed my car also thought I needed to turn it off before I could put the code in, and told me that it definitely seemed broken.
Well, the problem was that it really just need to be reset again. Once you put numbers in, you don't really know where you're starting from, since there's no read-out. So, if a two numbers already got pushed a day ago, it doesn't just automatically forget those two numbers after a day, unless the car's been on for an hour or more. If 3 and 5 got pushed yesterday, and your number is 11336... if you go to put that in, it'll read it as "35113", and then start over again at "36...". Even though two of those numbers were pushed a day prior. Hopefully that makes sense. So to reset it, you either have to leave your radio on (the car doesn't have to be running, just power to the radio) for over an hour, or disconnect the battery cable entirely for a minute or so, connect it back up, and try the radio code again.
I actually had the wrong number the first time around. I found a 5-digit number (all digits 6 and under, even) on my manual AND on the dealer envelope it was in, and assumed that was the radio number. I reset the car by leaving it on for an hour, tried the number again, and it still didn't work. Then I scoured all the papers in the dealer envelope that the previous owner had left in there, and finally found a "Pre-Delivery Inspection" sheet that had a "radio #" written on it. The same code I had been trying to use for the radio was actually a "stock #" and unrelated to the damn radio. I tried entering the radio #, but it didn't work. I reset the radio AGAIN (left it on for an hour), and FINALLY!!!! VICTORY!!!
So if there is anyone else out there scratching their heads about the trick involving turning the radio off, or "ERR" messages that you actually are NOT getting, then you probably have the same radio that I do. Just reset it and try again. The two sites I linked above have good tricks and tips about finding the radio number in the first place.
That was a hell of a lot more frustrating that it needed to be. I've always loved my Hondas, but THAT was ridiculous.