Yeah, women-specific bags DO make a difference, Jessie, and they're worth looking into. They aren't hugely different from "unisex" bags, but they typically have a few key changes. Mainly they're tailored to fit women bettera little wider in the hips, a little narrower in the shoulder. That makes them more comfortable to sleep in, but also improves thermal efficiency because the air spaces inside the bag have fewer gaps and pockets that you have to heat up.
Most makers of women's bags also bill them as having somewhat different insulation distribution, with more insulation around the torso and feet, but I'm not sure that makes a ton of difference.
Sierra Designs was one of the pioneers of women's bags. These days you might take a look at their Winema +0 bag ($279). Rated to 0 degrees, its also an environmentally friendly bag, with a lining made from Cocona fabric (a treatment that's derived from coconut shells but feels like soft nylon). It also has recycled PrimaLoft insulation. Weight is okay for a winter bag at three pounds, 12 ounces.
You also can get a down bag for your budget. Marmot's Women's Teton +0 ($269) has 600-fill down, insulation that usually is softer, lighter, and more durable than synethetic. And the Teton is a touch lighter than the Winema at three pounds, five ounces. It'll also compress more readily. A nice bag for the money.
Finally, in the bargain category, take a look at L.L. Beans Katahdin Climashield Women's Mummy ($149). It's rated to 0 degrees as well, weighs in at three pounds, five ounces despite having synthetic insulation (Climashield HL), and is highly rated by users. Worth considering!
For cold-weather camping, you need to also make sure you have a good sleeping padmaybe even twoto ensure you dont get cold through contact with the ground. Hat, gloves, socks, a snack of chocolate at bedtimeall can help keep you warmer.