I feel awkward addressing myself as Dr. X or Prof X. I know that this is common practice (e.g. in emails, letters, etc..). In emails, I prefer to just use my initials or first name. Is this a common experience (i.e. not wanting to be addressed as Dr. or Professor)?
Use whatever you feel comfortable with. That said, you should also tailor it to the type of correspondence when necessary. A formal letter should include a formal address for yourself, and less formal can be more or less whatever you'd like. In dealing with students, I generally sign school related emails as "Dr. G.," (with just the first letter of my last name) but I've also taught classes where I've gone by my first name exclusively, and let the students decide how to address me, after telling them it is fine with me if they use my first name. More formal student correspondence (e.g., if I'm cc'ing another instructor, or if the matter involves the administration) gets my full last name, and correspondence to colleagues is almost always my first name.
I also have a signature block, that includes my title, full name, email address and (sometimes) my telephone number.
I've been using my initials as the "sign-off" for my emails as long as I've had my email account. The difference between a formal email and an informal one is whether or not I include the "name block" or if I leave it out, which greeting I use ("Dear Prof. X" or "Dear Y"), and which valediction I use ("Sincerely" versus "Thanks" or "Cheers" or the like).
Normally, I don't know many people who sign their full names to an email. For an official letter, however, your address should be your full name, unless you know the recipient well enough to be on a first name basis with them. (And even then you might still opt for the full name!)
There seems to be two questions/issues here.
A lot of people I know do not use Dr/Prof/PhD when referring to themselves and in email signatures. I would say that this is quite common and perfectly acceptable.
The final statement you make is "not wanting to be addressed as Dr. or Professor." I think this is quite a bit rarer. I know a lot of people who tell students that they can call them by first name or Dr, or whatever they are comfortable with. I also know people who say "please call me by by first name". I don't know anyone who ever says "please don't call me Dr. or Professor." I would go so far as to say that demanding someone not use an honorific, or being offended when they do, is uncommon and not proper.
This answer is US-centric and based on my own experiences, so take it with a grain of salt.
"Professor" is a job title; "Doctor" is an academic title. Unless you're dealing with your students (in or outside of class), it's usually not necessary to refer yourself or to be referred to as "Prof. Lastname." It's common, at least in the United States, for anyone from grad students on up (and occasionally undergrads) to refer to each other by their first names, even over email.
A signature block is a good way to get around the issue; something like:
Dr. Firstname Lastname Florple Professor of Theoretical Blorplonomics Foobarbaz University, Room 12-345, (555)-123-4567
conveys your rank and title without having to worry about whether you should address yourself as Dr. Prof. X, Prof. X, Dr. X, etc. (In most situations, at least in science in the United States, being a professor implies that you have a doctorate, so the "Dr." above isn't necessary. The same line of thought about the email signature applies to doctors outside of academia, though.)
Personally, I generally just use my name, without any titles, except when (a) I'm in a formal or professional setting; (b) most of the people I'm around don't have similar titles; and (c) it matters professionally that I have such titles. Those situations don't occur very often.
I think that it depends on type of email or letter you are writing.
When you are writing an official email, it is better to use your initial as
Dr., but when you are writing to a friend I prefer not to use
Prof.. However, It all depends on the person. Some people prefer to be called
Professor Doctor Name Surname even by their close friends and some other are nicer and using their given name is sufficient for them.