A.B.
I think you're doing great! I like the prizes for being clean and dry--maybe I should try that. I am also potty-training my 3 1/2-year-old son. He is very different from my daughter who was completely trained by age 3. I am not using pull-ups because his school doesn't use them for potty training. You are right--they are like diapers. I only have them for naps and nighttime right now. But I do have to physically take him to the potty when he first wakes up. He wakes up too groggy to admit that he needs to go potty. He protests, but he goes. A lot, because sometimes he actually wakes up dry.
When we'd go out, I pretty much only went to places that had hard floors, and after a week, he was doing real well telling me he had to go. Now I can go anywhere. Pooping is a different story, but as long as they're not sick, it doesn't leak. Just bring plastic bags with you for the underwear after you've rinsed them in the bathroom. (And no, I don't throw underwear away like everyone tells me to--my son has yet to poop in the potty, and poops 2-3 times a day, and I can't afford to replace that much underwear!!)
Again, it sounds like you're doing great!! Good luck.