نَقَلَهَا وَهُوَ يَتَثَبَّتُ مِنْ عَدَدِهَا.
He moved them while checking their number.
نَقَلَهَا وَتَثَبَّتَ مِنْ عَدَدِهَا.
He moved them and checked their number.
In the first, “wa-huwa yatathabbatu” is a nominal circumstantial clause: the pronoun links the circumstance to its subject, and the wāw introduces that circumstance. In the second, a separate past-tense verb is coordinated, which does not equally foreground simultaneity.
The conjunction cannot be classified just by spotting a wāw. The grammatical structure following it and the intended relation determine the reading.