Welcome to the world of the stroller-free. I bet you will LOVE it!
I, too, would lug a lot of stuff around when we used a stroller. But I will tell you that each trip since then I have brought less and less stuff with us and I LOVE it!!!
Here's what we do:
Bought a SMALL backpack (maybe 12 - 16 inches high) for use specifically for WDW. IN this goes some granola bars (if not on the dining plan - no food if on the dining plan), ponchos (folded really small and ONLY brought when really necessary...If it's 90 degrees out we don't mind getting wet because it's refreshing), ONE small autograph book (not one per kid, one total), sunscreen and dh's glasses case (so he can switch between his sunglasses and regular). 1 bottle of water goes in the holder on the outside of this backpack.
Dh and I take turns with the backpack...It is REALLY LIGHT when we don't have the ponchos (which is 75% of the time).
In a cross body bag that I have on me at all times I keep money, tickets, camera.
We NEVER carry resort mugs - they aren't big enough to last much after the bus ride - then they're just dead weight the rest of the day. So I suggest leaving those in your room.
That's it.
That would be my best advice - Go down to the BARE BONES for what you bring with you and you will be much happier.
At WDW - you can bring a bag (small to medium at least) on EVERY ride. So no fears there.