Does it transform DNA? Sure. Why not. Its your book, you can really do anything you want to. It doesn't matter if it makes sense or not.
If you're looking into science based ways to do transformation, you can make educated decisions as to whether the DNA transforms or not, but it's really up to you.
DNA is what we build proteins from. We upregulate genes, and it causes us to make proteins that we use to manage our cellular lives. That's its entire job. Nothing more. Whether DNA is changed or not depends on how you want to have your transformation process to work.
You could make an argument that the DNA is changed to be more canine, then something kicks the cells into hyper-super-ohmygod-howtheheckisthispossible overddrive and instead of producing a reasonable amount of proteins per hour, they produce some ungodly high rate (multiple orders of magnitude faster than normal) to give you the raw materials to make a transformation occur. To have this process work, you'd have to alter the DNA.
On the other hand, with the amount of handwaving it takes to admit a science based human/animal transformation, you could also claim that this transformation occurs purely in the proteome (the collection of protiens in the body). This is just as scientifically reasonable as doing it with DNA, give or take, so there's no particular reason DNA has to get involved.
It all comes down to what you believe you can convince your readers is acceptable. Work done in the genome (DNA) has far stricter rules than work done in the proteome (proteins) which is actually still a relatively unexplored territory for science. If your readers want strict rules, DNA based transformation may make sense. If your readers don't care about the rules, but care about who kills whom and who hooks up with whom, trying to make a DNA based transformation process might get in the way of telling your story.
In the end, trying to explain how major physical changes (i.e. vampire teeth or werewolf musculature) occurs is substantially more important than whether a particular genome sequence is altered. Those changes are completely unexplained by either changes in the genome or proteome. Whatever reason you have for those changes to occur will completely dwarf any molecular level decisions you choose to associate with the transformation. You may choose to incorporate genome or proteome rationales for some of the changes, but 99.9999% of the changes are well beyond science's perception of what cellular mechanics of proteins and DNA are capable of achieving. Don't sweat the small stuff: choose DNA or no-DNA purely based on whatever makes your plot the simplest to tell, and never look back.
Remember Sanderson's first rule of magic: The authors ability to resolve conflict with magic is directly proportional to the readers understanding of it. As long as the understand what magic can and cannot do, it doesn't matter what technical approaches you assign to any particular magical ability.