I have a cute little cat that ran up to me when we got home, and it was rubbing our legs and crying....?

We can't have pets in our apt., but it's cold out. It doesn't seem feral or seem like it has rabies because it just cuddles up to us,then it tried to follow us in. When we didn't let it it started crying. I don't really want a cat. My fiancee says we could let it in for the night then take it to a shelter, but cats in shelters pretty much just get euthenized. I am wondering if it would be better just to let it run around and try to catch mice on its own. We are in Rhode Island and it is chilly out. What do you all think?