I think in spoken language 今朝 is perfectly acceptable and certainly not less natural than 今日の朝. 今日の朝 may be more colloquial than 今朝, but saying that 今朝 is "very formal" or "old-fashioned" seems to be taking it too far.
It might be a little far-fetched, but I think this is also reflected in corpus data such as 少納言. Even though it is a corpus of written Japanese, it does contain data from two online sites, namely Yahoo!知恵袋 and Yahoo!ブログ, where people are more likely to write as they speak.
Indeed, 50 out of 60 instances of
今日の朝 come from Yahoo!知恵袋 or Yahoo!ブログ. The remaining 10 instances come from books, at least 5 of which are direct speech!
50 online forum / blogs
5 books (prose) (?)
5 books (direct speech)
(I disregarded false positives such as 今日の朝日新聞, etc.)
As is to be expected from a corpus about written language, these are heavily outweighed by instances of 今朝:
1767 online forum / blogs
However, these numbers seem to suggest that 今日の朝 is more colloquial than 今朝, since books only use 今日の朝 0.6% of the time (in prose), whereas on online forum / blogs 2.8% of the time.