“Anglo-Saxon” is a bit of a broad term… you might describe Modern English as Mercian (an Anglian dialect), with lots of Saxon, Old Norse, and Old Norman influences. As well as other influences; I think those are the big ones.

Just to give an example of why “Anglo-Saxon” might be too broad a term—surviving Old English texts like Beowulf are in what you might call Late West Saxon, and the differences between Saxon and Anglian dialects help explain why Chaucer, who spoke something you might call Anglo-Norman, is much more accessible to modern readers.

If you want more of that Saxon influence, listen to West Country English, where it survives today.