SQL Always on Availability LSN Query
As I understand, you are performing a full backup and log backups on server 1, then you perform a fail over to server 2.

When you start performing the log backups on server 2 it follows the log chain.

So to answer your question: No it is not needed to perform a full backup job every time your availability group fails over.

I suggest that you test this in a pre-production environment. Do full backup on server 1, perform log backups on server 1, fail over, do log backup on server 2. Restore on another server the full backup and the log backups from server 1 followed with the log backups from server 2.

Good luck.

Stan