While it may seem like a nuisance to carry a bag with you everywhere you go, you’ll be glad to keep your hands free as you buy souvenirs and food. It will also make carrying extra layers much more pleasant. A daypack will be easier on your back than a messenger bag. A large purse may do the job, but it will also strain your arm and may draw the attention of petty criminals.
A small backpack is more than just a simple convenience. For me, it’s a necessity. I always bring a packable daypack with me. It takes up no room in my luggage and allows me to have a bag that is just for walking around. I got mine at EMS but I’m sure you’re smart enough to find one of your own.