i agree with the two bag approach- one messenger style left on the stroller, one small cross-body bag with your valuables (phone, money, credit card, camera, lipgloss, KTW).
I used a skip hop for the messenger, which worked out fine- you could go with a bigger model since you have two young ones.
My valuables were in a LeSportsac (I want to say it's called the Casey) which I highly recommend. Baggalini bags are just not my style- but the Le sportsacs are cute and very light weight, with a few zippered compartments to keep you organized.