Better than a "snow cone", there is a new Shaved Ice stand in DCA (Hollywood area, next to the "Avengers" stage).
Several interesting flavor combinations, and vanilla ice cream can be added (unfortunately, no red bean).
For your standard ice cream, there's Gibson Girl Ice Cream Parlor inside DLR. Also, Clarabelle's inside DCA. A few other locations serve soft serve as well. If you're looking for a few options outside of the parks, across the street on Harbor you have Cold Stone and Creamistry. A bit down the street on Harbor there's also Joe's Italian Ice. I've never been but I have a few friends who swear by it and anytime I've driven past I've always seen a line. Hope this helps
The BEST place is Joe's Italian Ice. They have Philadelphia style water ice, but also FANTASTIC soft serve custard like ice cream. They also do shakes, sundaes, etc. It's on Harbor a couple miles down towards Garden Grove.
You can get Snow Cones for the summer at DCA. They are part of the special Summer of Heroes food offerings. They are in Hollywood land.
There is a Ghirardelli Ice Cream shop in DCA in Fisherman's Wharf.
You can get soft serve ice cream on Paradise Pier or at the Cozy Cones in DCA. Also in Toontown, I think, at Disneyland.
If you want Hot Fudge sundaes, Ghirardelli is my favorite in DCA. I normally don't even care for Hot Fudge, but these I have to get anytime there is a Ghirardelli around.
Pretty sure they would have shakes too.