I have messaged support about this but wanted to give it more urgent attention.
I noticed that after I log out, my profile is still shown as Online when I check my profile page. In the first few minutes after I log out, it shows that I am offline, but after a while it shows that I am Online again. I have tried clearing my cache and history as well. This looks like a new issue. People may message me thinking I am online and ignoring them, when actually I have already logged out.
I changed my profile link to the new customized link, not sure if that has anything to do with it.