Anglo-Saxon dialects definitely existed, and were likely mutually intelligible. Historians can see variation from one text to another, something that has been used perhaps most extensively with Beowulf. Using the language of Beowulf, a compelling case has been made for a West Saxon origin, based on linguistic aspects of the poem, as well as some place-name evidence. Of course, Beowulf is interesting in and of itself, since the only extant manuscript of it, from the Cotton library, is a transcription from an earlier manuscript, and thus contains errors and changes from whatever archetype was used. Michael Lapidge has done a good job of surveying what we know about these changes in an article I'll cite below.

It's important to note first off, though, that the study Old English dialects is not a new field, and in fact dates to the late seventeenth and early eighteenth century. Language is an important aspect of national identity, and these early scholars of Old English such as George Hickes created a chronology of "pure" Germanic Saxon steadily diverging and being corrupted over time by Briton, Danish, and (outside of our period here) Norman influences. The idea of linguistic complexity was rooted in ideas of classical Latin purity, that the more singular a language was, the better. Of course, historiography has progressed since Hickes, but it's important to recognise that to some degree looking at Old English dialects is still an act of historical construction. Based on things like manuscript glosses and Old English texts, we've constructed an idea of the diversity of language in Anglo-Saxon England, but this is the written word, and may not reflect a potentially greater diversity in spoken English of the time. Nonetheless, these pieces of textual evidence do allow the differentiation of regional dialects. There was clearly a West Saxon dialect, as mentioned with Beowulf, as well as a Mercian, Anglian, Kentish, and Northumbrian. Certainly there were grey areas between the heartlands of these dialects, something that is picked up in documents like Beowulf, which shows aspects of the first four, but most strongly West Saxon and Mercian, hinting at an origin in northern Wessex or southern Mercia.

As to whether they were mutually intelligible, it's not something that will ever be proven, but is almost certainly the case. Although there are enough differences in the written word to identify one dialect from another, and indeed aspects of several within one text, they are similar enough from one another that Anglo-Saxons could probably communicate with one another regardless of origin.

It's important to mention too that I'm try to speak roughly in reference to the date you gave: c.700 AD. There is a chronology to Old English dialects, as with any language, and they evolved over time, to say nothing of the impact of Scandinavian languages from the Viking Age onwards.

Some sources/further reading:

Michael Lapidge's 'The Archetype of Beowulf'

George Hickes and the "Invention" of the Old English dialects', C.M. Cain, The Review of English Studies 11/2010

'Negative Contractions and Old English Dialects', L. van Bergen, Neuphilologische Mitteilungen 2008