To make the argument, you would substitute "thesis" & "antithesis" for "being" and "nothing" - like this:This form of argumentation that falsely presupposes the absolute separation of being and nothing, and insists on it, should be called not dialectic but sophistry. For sophistry is an argumentation derived from a baseless presupposition rashly accepted without critique; but we call dialectic the higher rational movement in which these, being and nothing, apparently utterly separated, pass over into each other on their own, by virtue of what they are, and the presupposition sublates itself. It is the dialectical immanent nature of being and nothing themselves to manifest their unity, which is becoming, as their truth.
In the next paragraph he says: "Becoming is the unseparatedness of being and nothing, not the unity ["synthesis"] that abstracts from being and nothing". Becoming is their "vanishing": "[t]hey sink from their initially represented self-subsistence into moments which are still distinguished but at the same time sublated."The form of argumentation that falsely presupposes the absolute separation of thesis and antithesis, and insists on it, should be called not dialectic but sophistry.
Does that sound like what people mean when they say becoming is the "synthesis" of being and nothing?
As for dialectic being a "method", in the Encyclopedia logic Hegel says (in the lecture notes to §41:
"it is to be regarded not as something brought to bear on thought-determinations from outside of them, but instead as immanent in them."