asking for iCloud account name and password to play iTunes radio when I have done it in the past many times without this question. Will not accept my apple id and password. Will not save my changed password. Where do I start?

I have a MAC with OS X Yosemite, version 10.10.2, 8GB, 2.7 GHz, intel core i5. Until 2 days ago, I could click on iTunes radio, pick my station and listen. Then 2 days ago, I do this and system asks for my iCloud account id and password. Does not recognize apple id or password. I change the password, does not save it. Screen that lists 2 step verification asks for added security questions, I do those and still will not allow me access. What do I do and where do I start? I thought I should download the update to OS X. I did that, still no change. Please help direct me to where I need to go to fix this problem.

Yogiowl