I keep getting alerted regarding unread Facebook messages even though there is none
It looks like Friend Caster is considered end of life so that may be the problem. Facebook apparently made changes to their api, so it has been discontinued. Source https://support.onelouder.com/hc/en-us/articles/204558634-What-Happened-to-FriendCaster-

You should be able to have 1 account on the app and one on the preferred web browser, or you could go the route this page talks about https://android.gadgethacks.com/how-to/set-up-two-different-facebook-accounts-one-android-device-0161254/ 

There are probably other workarounds but these are the ones I found with a quick google search.