It's at the low end of normal weight. So, my guess is no, you aren't overweight. "Feeling" fat is different than being overweight. A therapist can help you if you feel fat. Diet only works if you are actually overweight, and it helps you lose weight. It doesn't always change how you feel.
How old are you? Definitely not overweight, assuming that you are over 8 years old. If you're young, you might feel larger than your friends if you hit puberty and developed adult curves before they did.
Feeling fat and being overweight are two different things.
I was always told that for a medium built woman, calculate this way: for 5' tall, 100 pounds; for every inch over 5', add 5 pounds. Of course this will vary by body type and muscle, but it's an easy ballpark measure.
For ex: I'm 5' 8" and have slender bones, so I'm going for 140 pounds.
Based on BMI, you're fine. Are you feeling bloated? I always feel fat when I'm bloated. Maybe watch your diet for things that are making you feel bloated. Try to cut back sodium if that's an issue, or dairy.
Other than that, I agree with the above poster who suggested maybe doing some things to firm up. You have a healthy body fat percentage so things like ab work, and strength training are going to be very visible quickly (unlike me who has a lot of fat to lose before my awesome abs are uncovered). That might also help you to feel slimmer.
You might want to keep a journal of how you feel about your body each day and what happened that day. Maybe there's something you don't realize is a trigger to making you feel badly about your body.
I'm also the same height but weigh a wopping 190.... so no absolutely not you are NOT over weight. You wouldn't be considered over weight at 130 so you are far from being considered over weight. At one point in my life I was 117 but I consider me at my best shape I weighed 125 but that was also right after boot camp and 117 before boot camp