Jessie is a girl name, a nickname for
Jessica.
Jesse is a boy name. I like it much better as a nn, for a girl, but if you don't like
Jessica (or any other names beginning with
Jess, like
Jessenia or
Jessamine or something) than it would be better as a full name. It's a lot more substantial than some other nns. It's fine to me, and I love
Jesse for a boy.