We started sitters when our daughter was very young and had a teen sitter every week for date night. Lucky you to have family around!! We lucked out with a family who had 4 girls and everyone of those girls sat for us. As 1 left for college, a sister would step in.. It was great.
Make it an adventure for her to "show" her favorite toys to the sitter, play games, etc. She can "help" care for her baby brother.
I'm sure your cousin is qualified and will be able to redirect her if needed.
I personally am not a fan of distracting a child then running out the door. I don't like the message that sends to the child. I always told daughter goodbye but then again, this was routine for her.
Go enjoy yourself.. She is in a familiar environment with someone she knows... she'll be fine.