There are chatty expressions in English, like in the middle of a conversation someone says:
Does it work the same in German? as in
Or such a thing isn't used in German and sounds strange?
Do you mean oral conversation only or are you asking mainly about online chatting. Reason I ask is because you put "getting angry" between angle brackets, employed frequently by online chatters.
For both types of conversation, a "meta" comment about one's momentary feelings can be and does often get interspersed with the "regular" back-and-forth about the topic of the conversation.
Comic-book language, e.g,
Appeals to outsiders
Statements of personal status
To get back to your example,
strikes me more as a literal translation of "getting angry" than something that German speakers would actually type, or speak, as a meta comment in the midst of conversation.
on the other hand, could then be viewed as comic-book language (maximal and ungrammatical truncation of "Ich werde sauer") used to communicate one's personal status ("I am getting angry").
This particular example, in my opinion, would be found in online chatting sooner than in oral conversation.
There is such a thing, but it may still sound strange. ;-) (Depending on how much one is used to online chatting etc.)
It is called Inflektiv or Erikativ (after Erika Fuchs, translator of comic strips) and formed by dropping the infinitive ending -(e)n. Your example would thus become sauerwerd (also sauer werd, sauer-werd; official spelling rules don't cover it).
German Wikipedia has an article about Inflektiv.