A Suggestion
The web version of FS will automatically log you out when you log in on another device or browser. If the same held true when logging into the app, (i.e. logging into the app would log out the account on another device or browser and vice versa) I believe the sync error could be eliminated, and also the incidence of a new account being created by installing or opening the app could probably be minimized as well if we had to go ahead and log in each time the app is opened as well...
By no means do I know if this would work or is viable, but my old lady logic feels that it's worth mentioning...
As a young man who has been using the internet for entertainment since the year 1999, this is not the solution. Please do not create artificial restrictions that hurt the app's UX instead of actually making switching between browser and app "seamless".
I already have to deal with logining in a few times a day as I switch between my computer and laptop which is annoying, atleast for me. This is the last thing I would consider "seamless" or improvement to the UX of the app.