I'm severely allergic to dairy, and despite having been visiting the parks multiple times a year for the last 20+ years, I've only recently discovered all of the fantastic dairy-free options throughout the parks.
First of all, almost all QS restaurants in the parks have some allergy-friendly/dairy-free dessert. You can see on them on the Milk section of the "Allergy-Friendly" tab on
MDE's Mobile Order feature. Generally, it's a cookie(s) -- though some locations have a dairy-free brownie. For the "cookie" option, it's generally either a pair of small EnjoyLife chocolate chip cookies (which are delicious) or a larger cookie from David's (also very delicious, though a bit heavy for lunch in a theme park). Sometimes, the dessert is a bag of a bunch of "mini" EnjoyLife chocolate chips. It's all dependent on inventory, so there's no way to know in advance of picking up your order what you'll get. Even if you're not having a full lunch (just looking for a sweet dairy-free snack), you can place a Mobile Order for the Milk allergy-friendly dessert on MDE. It's worth noting that most resort QS don't offer these, just in the parks. Sometimes (though not always), the cookies are also available to purchase at the Main Street Confectionary, Zuri's Sweets Shop at DAK, and some of the resort gift shops. It's really hit-or-miss so there's no way to ever know for sure unless you check!
Several of the parks also offer dairy-free Tofutti ice cream (chocolate and vanilla), which is delicious. Availability is always dependent on inventory but it's usually available at Hollywood Scoops at DHS and Dino-Bite Snacks at DAK. It's also always available (along with a number of other dairy-free snacks like ice cream sandwiches and floats) at the Plaza Ice Cream Parlor in MK, though that is currently closed. I've heard that they may be currently offering it over at Tomorrowland Terrace (when that's open) in MK but I haven't been able to verify that firsthand. L'Artisan des Glaces in the France Pavilion at Epcot also has dairy-free sorbet. Some -- though not all -- TS restaurants also have Tofutti, Rice Dream, sorbet, and/or Divvies cookies available as "off-menu" desserts for those with allergies.
You mentioned Dole Whips. The pineapple flavor is the only one I've been able to verify as completely dairy-free. In terms of other common snacks, Disney's standard popcorn (the stuff at the kiosks and carts around the parks, not necessarily all the prepackaged stuff sold in the stores) is also dairy-free surprisingly. The flavoring is not butter but an artificial additive that is also dairy-free. The popular MinuteMaid frozen lemonade cups available at snack carts throughout the parks (as well as most resort QS) are also dairy-free. The Mickey pretzels are also supposedly dairy-free, though I haven't verified that myself. Some of the fruit-pops are also dairy-free but it depends which one. Of course, the cotton candy is also dairy-free.
You can always check at any QS location or snack cart and they can show you the ingredient/nutrition labels for the various snacks. You just need to ask and they'll bring out a binder with them all. While it's been a bit more hit or miss since the closure, Disney has also made an effort to label most menus (even at snack carts and kiosks) with a green leaf next to items that are plant-based, which genuinely indicates dairy-free.
One other option I have to mention is over at DS: Erin McKenna's Bakery NYC. Everything there is completely vegan (therefore dairy-free) and it's delicious! They have everything from cupcakes to brownies, so highly recommend grabbing a snack there if you're over at DS.
