Too many computers authorized, but I only have 5 devices showing

I'm trying to download some songs to my iMac. iTunes says I have to authorize this computer. (I've authorized it in the past and downloaded music to it before, but it's been about a year.) When I try to authorize it, I get a pop-up saying I already have 5 computers authorized. I go to my AppleID web page and it shows this iMac as one computer, plus my AppleTV, iPhone, iPad, and AppleWatch. Frustrated. I'm afraid my old MacBooks (which I've recycled through Apple and Gazelle) are somehow still on the list, even though I can see them under my AppleID. Really weird.