The phrase "also engrossed" is correct and usable in written English.
It can be used to indicate that someone is deeply involved or absorbed in something, in addition to another activity or state.
Example: "While reading the novel, I was also engrossed in the intricate details of the author's writing style."
Alternatives: "also captivated" or "also absorbed".
But it was also the year he met Käbi Laretei and Ingrid von Rosen, who would go on to become his fourth and fifth wives, while also being engrossed in an intense affair with Bibi Andersson, the lead actress in the year's two films.
She also became engrossed in eastern mysticism, music, yoga and meditation.
These special family traditions and expectations are both a blessing and a burden for Jewish women, who while lovingly connected to their cultural heritage are also deeply engrossed in a full secular life of work, exercise, relationships and reality TV.
I was also thoroughly engrossed with his command of the material, his methodical attack plan, the twinkle at every turn, and the barely-suppressed outrage he would have at something he thought was unjust or dishonorable.
As they have a greater ability to see the connection and utility of learning with regard to a certain phenomenon, they may also become deeply engrossed in learning about a topic in depth, without losing the big picture and while also seeing the interrelationships of other ideas to any topic of interest.
The extent to which bearing witness to their impossible choices diminishes Conrad's own life is salutary: Cartwright boldly contrasts lives of thought with lives of action, but is circumspect enough to note also the appalled, engrossed voyeurism that the fate of martyrs may inspire.
And Jacobi continued to publish the occasional piece in this vein also later, when engrossed in philosophical debate.
We also spend much time engrossed in our own tasks in our own workspace, so we don't have much opportunity to spread happiness.
