The phrase 'sitting there alone' is correct and can be used in written English.
For example, you could use it in a sentence like this: James sat there alone, lost in his thoughts.
She was sitting there alone.
"People might wonder, How strange that she is sitting there alone".
"They are just sitting there, alone, hoping someone will talk to them," Mr. Shige said.
On the other side of the ruined bridge, sitting there alone, was his son.
She was sitting there alone, and she made it obvious she wanted him to stay and talk.
On the other hand, even listening to her stories would have been better than sitting there alone, mourning my ruined life.
And he kept jumping up from the table, leaving me sitting there all alone.
They were each sitting there, all alone with their grief, not with a friend.
I saw no other guests sitting there, let alone over drinks, and I was forced to conclude that it was just a courtyard, one that felt like the world's loveliest and loneliest bus terminal waiting room.
The thought of sitting there all alone made me feel all panicky and sick.
I'm sitting there all alone, until finally some teenage girl and her mother show up.
