I just submitted my PhD application, how should I write an email informing the Professor about my application?
It is typical in the US that a student chooses a dissertation research advisor only after admission. Often it comes after a few years of advanced coursework that prepares the student to take comprehensive exams (qualifiers). Up until then, you may have an academic advisor (for advice) but without the fixed expectation that they will guide your dissertation. 

So, you have hope that this person will find a slot after you arrive. You should do what you can to get close to the person, though, through their coursework and (especially) seminars. Maybe something will open up for you. 

But you can also keep your eyes and ears open for others who would be good to guide your work during this initial period. 

Note that students entering with an MS and having already done sufficient coursework that they can take comps early have a shortened time scale, but probably still an opportunity after you arrive. It isn't essential, here, to have the final advisor on board at admission. 

Other places little of the above applies and one often enters the specific research group at admission. 



If you want to write the professor, pretty much anything will do if it just expresses your hope that you get a chance to work together in future. The emails you have received are entirely typical it the US.