Question: Whenever I try to backup a database it goes until 90% and gets stuck there until I manually kill (because it doesn't stop if I try to stop it) the msftesql process.

That clearly means that something makes a conflict between the fulltext indexing and the backup process. 

So, have you seen anything like this? If not, how would you go about debugging this problem?

Answer: 
What time do you happen to have the job that refreshes the Full text indexes running? Perhaps it is trying to repopulate those indesxes at the same time the backup is running.
I have the same problem.
The activity monitor shows that the Backup job has a wait type MSSEARCH

The index is manually populated when run it is hanging for days on end until I force-ably stop it or the service is restarted. it used to take minutes to populate.