• Charles-Edwards, Thomas M. (ed.). After Rome. Oxford: OUP, 2003. Short Oxford History of the British Isles: is perhaps the easiest read to have better understanding of the situations up to the 8th century, and it also has its Roman counterpart, The Roman Era: The British Isles: 55 BC - AD 410, ed. Peter Salway. Oxford: OUP, 2002.
• Fleming, Robin. Britain after Rome: The Fall and Rise 400 to 1070. Harmondsworth: Penguin, 2011 (2010): is an affordable as well as readable, and balanced account of the whole ASE. (Adds): Fleming has also just published the book titled as The Material Fall of Roman Britain, 300-525 CE, Philadelphia, PA: U of Penn Pr., 2021, though this latest monograph is not still published also in form of cheaper paperback edition (e-pub costs as the same as hardback edition).
• Higham, Nicholas & Martin J. Ryan. The Anglo-Saxon World. New Haven: Yale UP, 2013: is the more detailed overview book of ASE (than Fleming's), and generally recommendable, but it might include too much details as well as the chronologically arranged chapters might not suit your taste.
• Naismith, Rory. Early Medieval Britain. c. 500-1000. Cambridge: Cambridge UP, 2021: has also just been published. It is more detailed, but still very readable introductory overview work, and I suppose that the overall balance of individual fields of research (political, economic, cultural and religious.....) is about the same as Fleming's (though it is certainly more expensive).
• Oosthuizen, Susan. The Emergence of the English. Kalamazoo, MI: Arc Humanities, 2019: provides not a narrative sorted by chronology and various incidents, but a convenient summary of histriography (research history) on the transition from Roman Britain to Anglo-Saxon England, as well as some changing key concepts like ethnicity.

As for more recommended literature, the following previous posts might be interesting also to you:

• Did the Anglo-Saxons really exist?: answered by /u/VinceGchillin and /u/JosephRohrbach (especially the latter for additional recommended books).
• What are some good sources on/including early-medieval (c. 600) Anglo-Saxon England and Wales?: answered by /u/BRIStoneman and me.