First of all, breathe in and breathe out. Don't stress too much, this sounds like a fun assignment and you've already given it a lot of thought! Regina's music has been so important for me in getting acquainted with the English language, and I always enjoyed using examples from her work in assignments for high school/university. So I hope I can help you with this..
About the corpus: You mentioned both Songs and the song Mary Ann, so then I have to mention that Regina released a compilation CD at some point that is called Mary Ann Meets The Gravediggers And Other Short Stories. Which contains some songs from Songs (Oedipus, Prisoners, Consequence of Sounds, Daniel Cowman, Lacrimosa) but also some from 11:11 (including Mary Ann) and Soviet Kitsch.
I've been thinking about overarching themes and the first thing I thought about was capitalism and money. I think it comes back in many of her songs, including: Small Bill$, Trapper and the Furrier, Hotel Song ('a smile I've bought with a couple of gold coins'), Chemo Limo, Back of a Truck ( there's a back of a truck selling smoke free lungs / There's a back of a truck selling alien pods / There's a back of a truck selling game show hosts / There's a back of a truck selling the souls of the dead/ There's a back of a truck selling crumb free bread' etc. until they're even selling the back of someone's head!), Mermaid, Sellers of Flowers ('who's the winner? not the roses, not the sellers, not the buyers'), Scarecrow and Fungus ('he's got enough money to buy a new father'). And I can go on and on (and am tempted to keep going haha). And perhaps you could link this to her perspective of someone who moved from Russia to the US, a country of abudance where you could buy almost anything?
Just my two cents, hope it's of some use for you, and if I can think of other themes I'll write an update
I do think picking a specific topic or overarching theme will make it easier for you to structure the text. Like: Regina Spektor makes ordinary everyday things extraordinary in her songs, and here are three ways in which she does it (E.g. because rhetorical devices or vocal quirks make you think of a word or expression differently, or because the songs put a spin on familiar situations by highlighting new aspects).