No, you are not screwed, just might have to do like I had to do with my math, take non-credit prep course, they usually offer other subject prep courses.
It cost me more than I would have just had regular tuition but at least I was then ready to matriculate into the credit courses.
Don't give up, and do not worry if the college isn't super well known, after your first 4 years you can then go onto another college, transfer credits over to where you choose and are accepted.
Most often when you are transferring credits to a different college, there is not a big deal with acceptance compared to when you are fresh out of high school and applying to college the first time. Just my take on this.

My son is graduating HS this year too, and is going into psyc., also wants to be a psychologist and shoot for psyd.
he's even thinking of teaching elementary school afterwards or be a counselar.
At this point, who knows if he will change his major, but at least he is happy with this choice of his own.
For the first 4 years he is going to a local small college, nice student/professor ratio. He first wanted Drew University, but the tuition was outrageous (sadly,most are) and we, his parents, told him save that for your Psyd. or PhD.
At the college he will be going to he'll be able to do his Masters if he chooses.
My point I want to make to you is don't be self defeated, you'll do it, so even if you need college prep courses to be enrolled as a student, it's still worth the effort.

I was terrible student in HS, way back in the 70's, goofed off and had horrible grades, I went to college when I was in my early 40's.
No one is ever too old for college, ha! ha! In fact I am thinking of going back, get a bachelors in animal behaviour or psychology, who knows. Fist we have to concentrate on my son's tuition, thank God he has done fabulous with his grades, that helped get him a small scholarship.
You can even go part time to school, I did that cause of needing to be home most of the day, it took longer to get my degree, but was good for me especially with algebra and pre-calc, cause I needed to concentrate on that, other courses were easy.
Just do not give up, and do not give up on yourself

Sincerely,

DE