Construct a TM for checking if a set of parentheses are well formed
A very important fact about Turing machines is that there may be many equivalent Turing machines for the same language. In other words, just because your answer and the book's answer are different Turing machines, they may both be correct answers. There is not just one Turing machine for any given task.

I am not sure if your Turing machine is actually correct (building a Turing machine without any errors is hard!), but if you answer the questions in my comments I can probably figure it out and update this answer. We can also try a few more input strings to check.