When my kids were the age of yours I traveled with an umbrella stroller (even if they werent' in it I could put the backpack in it) a backpack and a small lightweight bag on a long adjustable shoulder strap.
I still use the small bag when we're in a city, amusement park even hiking. It's for my wallet or cash/ATM card, cellphone, camera, chapstick and sunglasses. Using the adjustable shoulder strap it can be made to be a fanny pack by shortening the strap - or a small purse with the strap across my chest / one shoulder (do you know what I'm talking about?) It provides easy access to cash and it's on my front so I don't have to worry about it being stolen.
The backpack is for sweaters, water bottles, change of clothes for the 2 yr old, diapers, wipes, bandaids, etc. My kids are teens so I rarely use a backpack anymore - maybe just a stringbackpack.
But I wouldn't leave home without the umbrella stroller. It's easy to drag around - it offers a place for your DD to sit when she's tired, even though there's no bag underneath you can often rig up a way to hand the packpack off the handles, or put a string backpack on the handles for loose stuff.
Just my opinion - but I wouldn't drag the big stroller and I wouldn't go with out one. Even with all kinds of family around - holding a 2 yr old can be exhausting - especially if it gets warm. And while the umbrella stroller doesn't have a canopy you can get a cute brimmed sunhat or even baseball hat for whish should shield alot of rays.
Good luck mama and have fun!