Mine is extremely similar to yours, made in 16 minutes last Saturday from scrap I had in the basement and a carpeted furniture dolly I've had for decades (Home Depot, about 20 bucks.)
Only difference in mine was I just screwed the 2x4s in from the bottom with two lag screws each, and I used scraps of 1x4 along the edges as lips, and didn't bother to pad them.
The doors pretty much stand up by themselves. In fact, I used just the dolly by itself a couple of times. You can stand the doors up and strap the tops together. But with the wooden poles it's more stable. Still strap them together with a nylon web strap and Fastex buckle from my backpacking closet.