Indeed, there is no contemporaneous record which can provide an answer to these questions of motivation from either side. And what retrospective assessments of the Anglo-Saxon migrations we have (Gildas and Bede) propose an invasion narrative which is viewed with great suspicion by modern scholars (as being archaeologically unsupported and inconsistent with the broader historical picture). So nor should the intentions and ideas of the participants, where they are asserted, be treated as amounting to anything other than the characterisation appropriate to the morality tale which suggested itself (particularly in the case of Gildas, who is literally sermonising).

It’s true that there are no Old Saxon or Old English records/accounts from the 5th century. These begin in the 7th century. But it’s likewise true that, particularly in the case of Old Saxon, when the written corpus begins to establish itself, extant (mostly liturgical) texts are of precisely the sort such as can do nothing to shine a light on this aspect of history.

As for when these languages and peoples began to diverge, I would rhetorically ask instead “when were they ever united, after all?” The readiness of the English to identify their language as merely “English”, despite by their own account migrating from different regions, to different newfound kingdoms, and speaking what are readily identifiable as differing dialects, is something of a source of fascination. But moreover, by asking “when did the Anglo-Saxons begin to diverge from the continental Ingvaeonic peoples”, we have to assert an implied initial unity of those peoples which is dubious at best. The Anglo-Saxons are linguistically closer to the Early Medieval Frisians (whom they do not claim a relationship to) than the Early Medieval continental Saxons (whom they do), after all. Of whom are we asserting a common ancestry, then, and why?

This having said, if Old Saxon was an Ingvaeonic language and we deem that a meaningful clade, it began to diverge from Anglo-Frisian, at latest, essentially the moment we have any record of it. This is mainly because the North Sea region which was its homeland was such a linguistic mess, within which mess Old Saxon was never the most influential (with Franconian and High German influences prevailing). Low Saxon (overwhelmed by Dutch and High German) and English (overwhelmed by North Germanic and French) compete for the status of most cladistically confused Germanic language. Frisian diverges from Old English much more subtly and minimally, and this divergence is on the part of English more than Frisian. Five centuries after their migrations began, an Anglo-Saxon would consider a speaker of Old Frisian to have nothing more than an unfamiliar accent (though the spelling system differs drastically, so this applies to writing not at all).