My First Big Screw-Up at Work

Started my new job a month ago as an admin assistant at this big firm. The team is all very friendly and professional, but there’s one guy, let's call him Alex, who always seems to be the center of attention because he knows everyone in every department. I didn’t have a chance to get close to anyone, mostly just stayed with my work trying to fit in. Today was when it happened. Received an email from our finance department asking for some documents that needed signatures by Friday or we’d incur late fees and penalties. It was all very urgent-sounding stuff that I knew had to be done as soon as possible to keep everything running smoothly. So, I started working on those emails right away. Had a couple of them ready before lunchtime, then decided to send the others over during my break since it was light work and there wasn’t much else going on around me at that moment. Typed up all these documents and sent them off with my boss's digital signature attached. Only problem is I forgot they weren't finalized drafts yet but were still in review stage with my manager. My boss had given the go-ahead last week, so I assumed it was okay without a second glance. Also, Alex was out for his lunch break when I did this, which would have been an easy time to ask him since he’s usually very helpful. When he got back, I thought everything was in order until my manager came looking for me right before the end of the day. He had a look that told me something wasn’t quite right and explained that these documents couldn't be sent out yet, not without his final approval. They’re now sitting on hold with all sorts of delays coming up. So here I am, feeling like an idiot who made a big mistake at my first proper job ever. Didn’t mean to cause any problems but really could have just double-checked things before sending them off. Everyone’s being understanding about it but I still feel pretty bad for messing this up so badly.