At this moment, it's very difficult to give a clear timeline, although the following information is generally correct (link):
Application periods generally last from June through October or November and, in some cases, January. Most schools admit in April while other schools also offer options to begin studies in the fall. In many cases, application timelines are the same regardless of your intended start date. Please check with the school you are interested in for specific dates.
As mentioned, every university has its own specific date for submitting the applications. There are universities that have a deadline of 1 November, others have 12 November, and others might last until the end of the month. Another problem is that the pandemic might cause the universities to make last-minute changes to the deadline dates, so you need to check their websites for news frequently. Lastly, not all universities allow for students to enter in the fall semester, so that is another thing that you need to double-check in advance with each university.
Edit: for most universities, you need to contact and secure an advisor before applying, but there are universities that don't need you to secure an advisor beforehand (in fact, some universities even forbid students from trying to email professors). It's unfortunate that the situation is so confusing, and this is one reason why most students start preparing the application 6 months or 1 year in advance. Keep checking those university websites in the next few days, you might find some that still accept applications (however, I admit that I don't know which ones). Also, keep sending your emails to professors, according to the template that you already prepared (its contents are fine). If the advisors do not respond to you, it will be because they are too busy or full of students, not because of your letter's contents.