Hi, DMahlon. Great question!
"Feudalism" is a problematic term for modern medievalists, as it tends to oversimplify a complex class of social arrangements that varied from place to place and from one period to another. It is also worth pointing out that "feudalism" wasn't a term used during the Middle Ages. Anglo-Saxon history extends from the 5th century until well after the Norman Conquest, so the social relationships of those people to one another changed drastically over the centuries, as you can imagine.
All that being said, the question of how social structures changed with the Norman Conquest is one that has been debated for the better part of a century. The two major historians that began this debate were J. Horace Round, arguing that feudalism was brought to the British Isles by the Normans, and F. W. Maitland, who argued that the Normans simply strengthened a social hierarchy that already existed. Neither position has won out yet, but here are some details that might interest you:
Immediately prior to the Norman Conquest, earls (the highest rank of nobility after that of the king) acted essentially as an extension of the king's power, ruling as his agent over vast tracts of land referred to as earldoms, which roughly comprised the boundaries of an earlier Anglo-Saxon kingdom, broadly speaking. After the Norman Conquest (though not immediately after), the earldoms were reduced in size to the shires or counties, governed almost exclusively by Norman earls.
William the Conqueror introduced a requirement of oaths of allegiance to the king from all those holding a feudal "office" within the kingdom, regardless of their place in the social hierarchy, in 1086.
Edward the Confessor reigned over England from 1042 to 1066, but he spent about 25 years prior to his reign in exile in the north of France, and some of his close advisers were Norman (Robert of Jumièges is the most notable). It is likely that, under Edward and prior to the Norman invasion, Norman influences began to creep into English society.
Even prior to Edwards reign, England was ruled by the Danish Jelling Dynasty from 1016 to 1042, and it is possible that Danish concepts of hierarchical rule mingled with those of the English, as is likely during the later Anglo-Saxon period.
In conclusion, someone who considered themselves to be Anglo-Saxon prior to the Norman Conquest may have called their contemporary systems of government Danish or Norman rather than Anglo-Saxon, but the evidence is incomplete, so this must remain an educated guess. What is known is that, following the Conquest, vassalages decreased in size and were conditional upon an oath of fealty to the king, who took a much more direct and prominent role in the government of his kingdom as a whole. Regional lords, who had been chiefly Anglo-Saxon under Edward the Confessor and Harold II, were replaced by Normans, while those on the bottom rungs of society remained predominantly Anglo-Saxon. But, as is often the case with history, getting more specific than that is difficult because of the ongoing debate previously mentioned, on which volumes have been written, and which can't really fit into this--already very long--reddit post. I hope this helps. Feel free to ask more specific questions.
Edits: Trying to figure out formatting.