This one comes up pretty briefly, but it's a major moment in the poem nonetheless. Baggage means a couple of things here: both the stuff she carried to the hospital and the kind of things you can't touch. You know how sometimes people talk about emotional issues as "baggage," as in "I can't deal with all her baggage?" Same thing here. The speaker wants to get rid of all her attachments to the world – the emotional connections that tie her down – and she uses the metaphor of "baggage" to talk about that feeling.
Line 18: Here's where she actually talks about baggage. While she's being partly literal, as she refers to her overnight case, we think she also might be talking about all the stuff that weighs her down, that stops her from being empty and free. She's happy that she has "lost herself" and the idea of having to hang onto anything makes her "sick."
Line 20: Now things get a little sadder. It's pretty clear that the speaker is using "baggage" as a metaphor for her connections to her family. Instead of being a source of comfort, they are a burden. That's no way to talk about her family, but by turning family into just another piece of luggage, an object like a leather suitcase, she conveys the depth of the change that is happening inside her.