This is admittedly coming from somebody who's "real name" isn't one that lends itself to being shortened or adapted into a nickname...  but, I say leave it as Jessica.    Also, just let them call you Jessica during the interview so as to avoid wasting a minute on your name.  Then, if you land the job, you can transition everyone to "Jessie" once you start working there.  Or, who knows, maybe you'll end up liking being "Jessica" in the work context.  I know an attorney that was "Kenny" for the first five years of practice, or so but is now trying to get everyone to start calling him "Ken" and it is very hard to break people of the habit.

My first question sounds silly, but how many people know you?  If an attorney you're interviewing with next week is out to lunch with another attorney who happens to be your biggest fan, will the first be able to realize that the "Jessie" the latter is going on about is actually the "Jessica" whose resume is on his desk?  Or if an attorney you're interviewing with has already met or worked with you in some other context, will he make the connection that this person on paper is the person he already knows? 

Personally, I would tell you to go for it.  If you know that you want to be called Jessie, put it out there.  It's not an unusual or unprofessional nickname like "Spike" or something.  Honestly too, I think there are a lot more people our age that have the nickname as their legal name.  I have a friend whose name is Christy.  Her birth certificate says Christy, not Christina or Christine.  If you put Jessie on your resume, no one will know that's not your birth name until you fill out your W-2s.

I think it depends on the nickname. "Jessie" doesn't sound unprofessional to me, but if you have "Jessie" on your resume, then I think you have to be prepared to be stuck with it (i.e. have "Jessie" on business cards, nameplates, email signatures, official correspondence, etc.). If Jessie feels like your name in the sense that you can't imagine ever going by anything else and are comfortable with that sort of professional permanence, then I don't think there's any reason not to introduce yourself as Jessie. On the other hand, to start out as a Jessie and to switch to a Jessica, or to be Jessie in some professional settings and Jessica in others, presents a wishy-washy image that may not hurt you directly but might infantilize you in the eyes of at least some co-workers. I think it's best to pick one name professionally and to stick with it, and if that name is not your nickname, then it will create a feeling of closeness between you and co-workers you become friendly with to let them transition to calling you by your "my friends call me..." name. I think this distance between your personal and professional self can be helpful, like not bringing your work home or your home to work, but I think it's a personal preference.