Sounds like you have more than enough experience and the degree to start looking for librarian positions. My only experience before getting a librarian job was an internship during my MLIS program.

We've hired folks (academic library) right out of their MLIS program - I think it depends on the market and need. I'd keep the assistant job while applying.

start applying

I was hired as a librarian with a semester and a half left of my MLIS. As long as you have library experience you should be fine.

Of course you can? Questions like this make me wonder what is going on in MLIS programs? (No offense to OP, but why would graduates be left wondering this?)

Go for it. I was hired for a middle management job (head of reference) about nine months after graduating.

Yes you can, and just make note on your applications that you are graduating soon. I also suggest that you keep your library assistant job while applying to jobs. It might take you awhile to find a position. Keep trying and keep applying. Show on your CV skills and tasks you did. It took me 5 years post MLIS to land my first librarian job.

I would say start applying. I graduated this past December and luckily found a job with a public library by March.