Postgres database insert become slow after 10 days, and then it needs a full vacuum
If performance is restored after a VACUUM ANALYZE, I would question whether the table(s) in question are being auto_vacuumed and auto_analyzed.
If you look at the pg_stat_user_tables view, do the values for last_autovacuum and last_autoanalyze seem reasonable?
If tables are constantly being locked then auto vacuum might not be able to run.