Is あたし seen as soft and girly like the particle わ is? Or can a girl just say it and not sound emotional or dramatic or soft?
Yes, あたし is a very feminine form of わたし.
However, あたし isn't really often used outside of anime and idol usage, as it gives a very unrealistic "cutesy" kind of vibe. (at least in my experience) Of course I'm sure it's used appropriately somewhere, but it's best to stick to わたし/わたくし for everyday usage.
EDIT:あたし can be used in real life applications as a softer/feminine わたし (thanks to broccoli forest)