Yes, it is awkward to start a letter or an email with an introductory statement of your name.
If you are writing a paper letter then your name address will be at the top of the letter; the structure of letters should be readily available somewhere on the Internet.
In an email you can discuss your circumstances (e.g. current job or school) in the body of the letter if and where it is relevant, but it is sufficient to place your name at the end, along with your address and other contact information.
For professional (or otherwise formal) emails it is helpful - I would even say even advisable - to get an email address with "firstname.lastname@example.org" structure, e.g. email@example.com. It looks more credible than firstname.lastname@example.org.
You'll find more guidance online, but avoid "I am writing to you to..."; it's obvious you're writing, you don't have to say it.
The best thing to do is to state the context, either in a separate "Ref: such-and-such a position" line, or in text: "In reference to the xyz position..."
And don't say "in regards to": it's "in regard to" if at all, and even that is better left out.
Best to use the the "Ref: xyz position" line, and start dialog with something like "I would like to ask some questions about this position.."