Live in New York, want to form DE LLC
First of all the answer is yes, you can form an LLC in Delaware, even though you live in New York.
However, with that being said you need to be aware of the fact that if you intend to hire employees in New York, open an office, and/or deduct expenses such as home office expenses, gas, meals, etc. - anything that would make strong connection between your business and NYS - then you would have to register this LLC in New York as a foreign entity.
So it might be a good idea to consider forming the company in New York.