Sounds a bit like a problem I had, signed up to flick on a whim, then forgot about it. On going back, I found I could not sign in. Went through the same rigmarole you are going through, ARGH! After several several more attempts, gave up again, in meantime I had read an article in Computer active about some advantages in using different browsers. I am on Firefox, and I had loaded Chrome, so signed into Flickr with Chrome, went straight in with the original password.
Still can't sign in to Flickr with Firefox, just get the wheel of eternity. Still going through a load of problems with Win. 10 May update so not too worried sorting out the differences between Firefox and Chrome and as I can't find the article in Computer Active I am not too sure what the advantages of using several different browsers are.