Ok, I have a question, because I am not sure how to handle this situation. Me and this girl have been seeing each other for a few months. We had been friends for a couple of years beforehand, but circumstances would not let us be together (we lived a long distance apart), during the time we were apart, we did not talk to each other and lived our own lives.

Well, zoom forward to the present, we started living in the same city again and have started seeing each other. Everything is going great, except for the fact that her job prevents us from spending that much time together, however, we really like each other and have exchanged several "I love you's". Initiated by both parties separately. She currently is on a business trip for a couple of weeks (one of the reasons we cannot hang out as much as we both want), we had been talking at the beginning of the trip, but she began to become upset at work and was really depressed about how it the trip was going. She started telling me about how she was getting depressed, unfortunately, I was out with some of my friends and was drunk when we were talking... it seems to have had a bad effect, because ever since then she has completely shut down. She still talks to me, but it is not the same as it was before (she used to tell me how she missed me and wanted to see me) She has told me that she misses me, but that when she get really depressed she completely shuts down. I have apologized about the drunk night and told her that if she needs to talk to anyone, that she can always talk to me. I feel like I really dropped the ball that night with my friends. I have been giving her some space, I just don't know if there is anything else I should be doing? Any ideas? Continue giving her space? Try harder to talk to her? Send her flowers?

If you're trying, and she doesn't want to talk that's all you can do.
I'd call. Tell her you love and miss her, but you feel that she's being really withdrawn and your not sure how she feels so you'll leave the ball in her court.
See what she does...

Send her some flowers and tell her you miss her then give her space. She told you she shuts down, it sounds like what she is doing.

There is never a bad time to send someone flowers. Well okay maybe if they are allergic to them, that would be a bad time. But still, they do work.

Continue the good communication with her and stop beating yourself up for "dropping the ball". There is nothing wrong with you going out with friends.

It's time to do something special for her. Just a quick note and some flowers or a pretty plant delivered to her saying you are thinking about her. If she responds, ask her out for coffee when she returns from her business trip.