As others have said, less is more. Paperwork, phone, charger, wallet. Compression shorts and sports bras, if so inclined. As for your own running shoes: absolutely, if you can swing it.
Go to a reputable running store with knowledgeable people. Have them check your gait and determine your amount of pronation. Then follow their reco for the right pair of shoes. Forget color or design or brand -- this is all about comfortable efficiency. Buy a pair and run in them prior to plebe summer. Bring two pairs if you can. You won't regret it.