I changed my passord and now it keeps signing me out!

I have the iPhone 7+ and I got an email that someone was trying to access my information, so I changed my password. Once I did that, the password kept failing on me and I have to keep signing in! Not only that, but even though I reset my password, it accepts the OLD password instead of the new one, and yet online it accepts only the NEW one. I'm so confused why it's being so freaking stupid and it's really annoying me. I can barely use my iwatch anymore because it stops working when the iPhone randomly signs out of its password!!!!