I'd like to preface by saying I am not a consultant, this is just tips from my personal life experience! :)
To help with getting the wax off your doggy, have you tried coconut oil? Just rub it on the coat around where the (dried) wax is and then it should come off relatively easily. After that, it's safe for your doggy to eat (and it's even good for their coat, win win!)
Second, I'm assuming you have a plug in warmer, based on your story. I have one as well, and a few mid size, and I much prefer the mid size. HOWEVER, if you don't want to buy those little tabs to make them easy to get out, you could just wait for the wax to harden (assuming you're changing it after it's been on for awhile, if not, well you don't have to wait!) and then turn the warmer on for about five minutes. After that, just a small amount will have melted, and you can slide it right out! I've found that much more safe than trying to get the wax out after it's all melted, and using a knife to get it out is just annoying usually. That's the best tip I have for you. Good luck!