Old English had some mutual intelligibility with other Germanic languages such as Old Norse and Old Frisian, however this doesn't necessarily follow that they would then adhere to some sort of "Germanic" identity that would compel them to return to their long abandoned homelands to reconquer those areas.

Identity in Anglo-Saxon England is quite an interesting topic with a great deal of variety. There were numerous and often overlapping identities that were at play. The Venerable Bede tells us that many of the Anglo-Saxon kingdoms claimed descent from certain parts of northern Germany and Scandinavia (we know in reality there were no such neat divisions), but there were often far more important markers than supposed ethnicity. Religious identity, as Christians (which we would call Catholic today) was of primary importance, and later on for example in the Viking Age the English/Anglo-Saxons were often contrasted with the "Danes" indicating at least some level of "othering" going on between the two.

But this does not mean that the English were not "Germanic", whatever that means. Many pre-Christian stories and artistic themes carried over into Christian times for example, artworks such as the Frank's casket contained both scenes from Christian and "Germanic" mythologies despite having been produced well after conversion. Literature was likewise influenced by both, looking at you Beowulf. English kings, and later Kings of England, still were "Germanic" in that they spoke a Germanic language, patronized figures such as scops (similar to Norse skalds), were likely familiar with pre-Christian myths and stories. However they would not have understood themselves as "Germanics". "Germanic" is a classifying tool for linguists, anthropologists, etc... Things that are evident to them, such as shared linguistic/cultural ancestry are not necessarily evident to people on the ground in historical contexts.

To sum up, there were aspects of "Germanic" culture that the Anglo-Saxons were familiar with, but this doesn't mean that they actively identified with Germanic culture and sought to place themselves within that context. It is more useful to think of this as a cultural substratum upon which other things were built.

As for their relationship with old Saxony....we don't have much evidence. During the migration period itself there were not organized polities as we are used to envisioning to carry on relations between the two areas. Later on as organized polities arose in post-Roman Britain, the continental Saxons were increasingly drawn away from the North Sea area of activity, and instead drawn into Frankish orbit. This was a more riparian system, operating North to South along rivers such as the Rhine, and also increasingly centered on Francia, specifically what became West Francia, or roughly modern day France. The various Saxon homelands were subjugated, somewhat violently, by Charlemagne in the late 8th century, more or less ending their interactions with the English kingdoms. (Though traders from both Francia and Frisia were quite prevalent in England).