No. These days EU citizens get no card at all, they just get a certificate confirming their right to reside in Poland. It is good for five years and there is no penalty at all for not having one (so why bother).
Yes, she is Polish. She's wary of living in another country, because she's finishing a degree on Slavic studies, and she's not sure how easy it will be to find employment abroad with that. But, if we find it too difficult to make it in Poland,
OK, I've had a think and a couple of chats and it looks like you won't have too many problems. You can either apply for a residency permit (i.e. karta pobytu, a temporary resident's card) on the basis of a job in Poland (easier to get but often the card will only be valid for as long as your work permit is valid for, which means that you have to get a new card every 9 to 12 months) or on the basis of wanting to reside here with your unmarried partner (admitted the only people I know who applied on that basis where hetero but in theory there should be no difference, the regulations are the same). The downside of applying for a card on the basis of wanting to reside here in order to be with your partner is that you will need to prove that you have sufficient funds to support yourself throughout your stay. That can be cash in the bank or it can be a commitment from somebody to supply you with a guaranteed amount of money per month. The upside of it is that your card will be good for two years.
Even if you are married, you still need to have a resident's card. I believe that the first card is for two years and then every card is for five years (but I could well be wrong on that, I'm not non-EU and I'm not married either). The only advantage is that you do not need a work permit (if you aren't married you need a different work permit for every job). It might be interesting for you to get married in an EU state and then try to get married person status in Poland (i.e. exemption from the requirement to have a work permit). I'd certainly like to see what happens!
I suppose that as an absolute last resort, you do have an American passport: I doubt it would be much of a problem to arrange a paper marriage with a Polish man who wants to get an American passport in exchange for you getting a Polish passport.