Most insurers won't cover pre-existing conditions. So if Waffles has had any illnesses, even something you haven't claimed against from AXA, a new insurer will want to know about them (and may check with your vet) and probably won't provide cover.
It's not only major illnesses and conditions. As an example, some insurers might not provide cover for any future ear problems if your dog has had perhaps 1 simple ear infection that cleared up very quickly.
Have a look at the recent thread about AXA where somebody on here had a huge premium increase.