I've been walking dogs daily in the New Forest since 1987 and during that time I've seen adders probably a dozen or maybe fifteen times in all, mostly sunning themselves in a sunny spot fairly early in the day to warm up. They're active from about March to October and can be found on heathlands and in woodlands through much of the UK. The New Forest is just one of many places where they can be found.
I don't worry about them very much, but I make sure I know what to do just in case. The closest encounter any of my dogs has had was many years ago when my first 2 Cockers were trotting just ahead of me on a wide path in fairly open woodland one warm afternoon. I saw an adder coiled up on the path just as the dogs drew level with it. They didn't notice it and had trotted past it before I even had time to think what to do. I let the dogs carry on and walked past it myself, just giving it a bit of a wide berth.
In all those years of dog walking in the New Forest I've personally known just 2 dogs that were bitten, a Cocker and and an English Pointer. Both survived although the Pointer was very poorly for a few days. I've heard of several other dogs being bitten but I've only ever heard of 1 that died, which was a young Husky puppy.
So please don't become unduly worried. Any self respecting adder will slither away given the chance if it hears noisy dogs or humans approaching and if you are one of the very few unlucky people whose dog is bitten follow the good advice given by Hannah.
My vet practice, with surgeries in Lyndhurst and Totton will deal with emergencies for animals not registered with them. Their website is
http://www.midforestvets.co.uk/homepage1.htm and it might be worth checking their location as well as the one that Wendy posted the link to and see which is nearer to where you'll be staying.
PS. Don't do what one of my work colleagues did a few years ago. Despite being a local girl who should have known better she touched an adder that she saw sunning itself and ended up in hospital for a couple of days after it bit her finger.