2000 Honda Accord 3.0 L V6 – battery not charging

A few weeks ago, I was driving my accord and suddenly a bunch of lights illuminated on the dashboard, including all the doors, trunk, SRS, battery, and maybe others (some bulbs are blown). They stayed on a while and went back off. This happened several times over the course of a week or so until finally, they stayed on. I continued driving it, and started realizing that the battery wasn’t charging. I wasn’t sold on it being a bad alternator, but I went to the parts store and talked to a guy there who had Hondas and he said that they would often do odd things like this when the alternator went out, so I bought a new alternator and put it on. It acted normally for a day or two and started doing the same thing again. I had a lot going on the last few weeks and the car has just sat and now the battery is dead. I jumped it today off my other Accord. All the lights are still illuminated on the dash and it dies as soon as I unhook the jumper cables. I find it odd that the alternator would be bad since I just replaced it. I sort of thought from the beginning that it was computer related since the symptoms would come and go. Any thoughts? Please advise. This car has over 400,000 miles on it, and I want to keep it going without spending loads of money on it.