I should be okay to stay in the country for a month without any sort of paperwork.
That is true. But you will need to apply for your residency permit within an absolute maximum of 45 days of arriving (you're best off getting it in 30 or 31 days after you arrive).
I can just go to the American Embassy and apply
That's complete and utter bollocks. The American embassy have nothing to do with the process of becoming a legal resident of Poland. The fact that they have told you that is very worrying: either they have no clue what they are doing or they simply are not going to bother getting you legal in Poland.
And I'll make net 1807zl a month for the first three, then I get a raise.
Your life is going to suck. A two-room (i.e. one bedroom and one living room) flat in Krakow in a neighbourhood within about 30 minutes of the city centre is going to be 1600zl plus bills (200zl per month), so you're looking at 900zl each if you share. Add on 50zl each for internet, 50zl for your cellphone and 100zl for your tram & bus pass and you're up to 1100zl before you've bought any food or other consumables. Add on 20zl per day for food (which is going to mean a lot potatoes and cabbage but very little named meat) and you're at 1700zl. I don't know many women who spend just 100zl per month on sanitary/cosmetic products (soap, shower gel, shampoo, hair care products, tampons, toilet paper, make up, lipstick, perfume), but you're going to have to. I hope you weren't planning to ever go out, travel, take a taxi buy any clothes or shoes, get sick or have money for a plane ticket home when you want to leave Poland.
you have to take a weekend trip to Ukraine and spend the night there, in order to give them more time to process everything.
That's rubbish. Going outside Poland does not
give you another 90 days inside the Schengen zone. And more to the point, if you got your application for a visa in on time, your existing visa is automatically extended until the foreigners office have finished processing your application.