How to fix CRC error when doing a SQL Server database backup?
I'm having is that I'm getting an error when trying to backup the database

That's potentially bad.

I have not done a Transaction Log Backup ... I'm in Full Recovery Model

That's actually good. Then backup the log.  Somewhere safe.

In my test server I restored the backup that I have from 3 weeks ago

Almost home.  On your test server restore that old full with NORECOVERY, then restore the log backup.  If that works, your data is safe.  And even if your database is corrupt you can restore from the old full, the big log backup, and a Tail Log backup from the production server.