I've played a lot in play by post games, and it was w very performative experience, especially since when I started I was an eighteen year old just after her first publication, so writing was more important for me than ever before. Pbp's were both my writing polygon and the way of seeking validation, so yes, I always had the audience in mind, and my posts were probably more flashy than they would otherwise.
I also played one-on-one sessions on google docs, and they were different, but not as much as I thought they would be - I still had a high need for beautiful language for example, and could polish the posts forever, but also I made narrative choices that I wouldn't do if I knew that anybody on the forum can read this.
On the other hand, when I got other source of validation, I started to come to pbp games with more loose attitude - I focus less on making the posts attractive and interesting to my co-players, more on getting what I want from the game (usually the feels ;) ).