If you need to clean her bum, you can use a warm damp cloth to get it off. If it's stuck on dry and hard, then just give her bum a bath. Don't soak her, just get her bottom wet to work the poo loose.
You can encourage more hay eating by refreshing the hay a couple times per day. Don't wait for it to be gone to refill it. Just put fresh hay on top twice per day. This usually gets them to eat more. The fiber can help prevent the messy bottom.
You are right about the carrots. When you do give her some on occasion, try to limit it to about a 1" slice and no more.
Many of us on the forum have gradually introduced greens to our rabbits and feed them 2-4 cups of greens daily. Mine get that daily and have never yet had poopy bums. Some bunnies react differently to specific greens, however, which is why greens are introduced one type at a time.