Hi Aux2,
I agree with people above me, you might have more chance than you think! I went to a tiny liberal arts college, and majored in psychology with a concentration in linguistics. I ended up applying in my fourth year right away for Linguistics PhD (in retrospect, if I knew how little background I have, maybe I wouldn't have, but I had no idea). My school did not have a linguistics program at all, and only hired a professor with a lingusitics phd in my second semester third year. That was when I took Syntax, fell in love with theoretical syntax, did a research project my senior year, and ended up applying to strictly theoretical programs. I decided to look into linguistics programs summer before senior year. What was I thinking...
Anyways, applied to 6 places, got into 2, attending one. Only realized how little background I have when found out that in my cohort of 6 people, only two of us are coming straight from undergrad (and I got funding while the other person didn't), everyone else already had a MA degree, and I was the only person with no linguistics major at all. I did not feel that my lack of background was much hindrance, there were people stressing out and working much much more on homework assignments than me in my graduate classes.
Let me know if you have any questions!