Is "I am writing to inform you" awkward or outdated?

I don't believe it outdated, but very much reserved to:

• business letter (where it is one of the common phrases: "I am writing to inquire about ...")
• application letter ("I am writing to apply for the position of ...")

Some guide on writing style will advise you to:

Avoid stage directions.
Do not commence a letter by telling the recipient what you plan to do in the letter or begin an essay by telling the reader what the paper will discuss.

"I am writing to you to ask you to ..."
"This paper is going to talk about how ..."

So if you only want to express the intent in a less formal way, a simple "I would like to inform you of ..." is enough.

As VonC at least implies, the basic construct is somewhat formal. But I think "I am writing..." is slightly informal by comparison with "I write..." - so for me, it sits uneasily with the more formal "...to inform you..."

Interestingly, Google Books bears this point out. The informal version of "...to inform you..." is "...to let you know..."; it turns out the pairing "I am writing to let you know..." (both informal) is much more common than "I write to let you know..." (mixed formal/informal) by 8,840 to 5,960.

On the other hand "I write to inform you..." (both "formal") is in fact more common (by 17,900 to 14,000) than OP's (mixed informal/formal) "I am writing to inform you...".

Personally, I think the formula doesn't have much to commend it. What's the point in starting a written communication with I write or I am writing? Obviously you are - in the fullness of time they'll be reading what you wrote, so they know that. And do they really need to be told that you're going to tell them something? Just get on and actually tell them whatever it is you want to say.

It is clearly redundant and circumspect. However, it sounds polite, and some situations do call for social niceties.

I am writing to inform you that I'm writing to inform you that I'm writing to inform you of certain errors.

vs.

Please address the following errors:

If I were to act on the information, I'd prefer the more direct style.