Sounds like maybe a cookie issue with Firefox - check the browser settings to see if any recent updates have updated the cookie preferences.

You can actually be signed into two different accounts with any browser that supports untracked tabs - eg with Chrome if you open up a new tab (or window) in incognito mode, you can log into a different account. If you close the tab or window you'll need to sign in again. 

Opera is a decent alternative to Firefox, too (I use Opera and Chrome after FF became unusable for me). It's not as popular so it's not as catered to as Chrome and Firefox, but mostly for me that just means I have to use Chrome when paying certain bills.