Steer clear of the US/Mexico Border, especially in Texas. Arizona can be nice during the winter, but don't go too far south, there's a lot of issues down on the US/Mexico border these days. (I was living in Arizona until recently, a stone's throw from the border, I experienced some of it first hand.) I'd also avoid the west coast states. Places like Oregon (Portland area especially), again, a lot of stuff going on out there. California isn't too bad in some places, just do your research on where you go and look into what's good and bad. There is safety in numbers as well.
Also, traveling alone, and being a female, carry a gun or two with you. And some pepper spray. Not sure how easy that will be coming from Quebec, but if you don't know guns, take a safety class and get yourself a pistol that you can handle. Me and my better half keep a gun, and other self defense weapons, in our RV always. P.S. Make sure you get the proper permits for the state you will be staying in. Some states are open carry (meaning no permits required) but its better to have some kind of a federal permit like a CCW.