My first suggestion is that you talk to your boss to figure out Jessica's role in the onboarding process. If you feel that will be perceived negatively, then give feedback directly to Jessica. have a non-confrontational verbal exchange with Jessica. Focus on removing barriers and minimizing defensiveness by approaching the situation, so Jessica feels she is helping you out of a difficult situation, instead of being a pain in the ass and causing you angst. Here's what I suggest to have a positive and productive dialogue.
Make a list of exactly what you need to learn about the position and what makes you l uncomfortable.
Tell Jessica to let you know when she has a few minutes to talk.
To get the conversation started, talk to Jessica, and thank her for her help. Say you don't know how much longer she's going to be around, or what her plans are, but you appreciate her help. Tell her you're excited about the job and want to perform well in your new role. Ask her what she found most challenging about the job when she worked there. You might ask her if she has any advice for you and if she does, listen and thank her.
Explain that you are coming along well and you're a quick study, so you feel quite confident about your new role. Then, you might segue to your list. Say that before you are totally on your own that you have a few questions, but you're uncertain whether to ask her or your boss since she no longer works there. Then, continue as appropriate.
During this exchange, you can get clarity about her role and find out what the heck is going on. Since Jessica is friends with your boss, this approach is wise, as it's likely to get back to her.
I hope this gives you some tips so you can craft specific talking points based on your everyday reality. Also, here are some tips to help you have a successful transition during the first 90 days of your new job.