You're right, X has a binomial distribution with parameters n and p.
You don't need to worry about the estimator p-hat as the distribution would be the same even if you weren't calculating an estimator.
If a question was something like this...
A poll taken of n=100 randomly selected people estimates a proportion of a large population that prefers Food A. The actual proportion of the population is p = 0.64 that prefers Food A. Let X and p-hat be the number and proportion respectively of the 100 polled who prefer Food A.
Now, the question asks to name the distribution of X and give the values of any parameters involved. I am not sure what distribution this is (although I suspect binomial) and I'm confused by parameters... help!