No, you can't. Both...and are not to be used together with 'don't. However, they go well with negative verbs like in: Both Ann and her mom dislike chocolate.
Neither she nor I knew what to expect.
not only..., but also:
She bought not only a new car but also a new lawnmower.
You may either have the ring or the bracelet.
They are called pair words since they go hand in hand, at all times. Also, grammatically they are called correlative conjunctions (i.e. related to one another).
Other ex.: scarcely...when, not only...but also and so on and so forth.