How homogeneous was Old English spelling?

Are varying spellings available, or was Old English rather uniform, as far as the sources show?

Variant spelling may have indicated different verbal dialects, but written dialects, involuntary eye dialect, may allow greater insight into the pronunciation. Is this cleaned up and normalized in OE dictionaries?

Old English spelling wasn't entirely uniform. I don't know how to judge the precise degree to which it was non-uniform. Old English words could have more than one spelling, and dictionary entries certainly make use of normalization: the most common normalizations are the use of symbols like ċ, ġ to represent palatal consonants and ā, ō, etc. to represent long vowels, the exclusive use of w instead of ƿ, and the use of only one of þ or ð.

The letters þ and ð both represented the same sound(s) in Old English (typically analyzed as a single phoneme /θ/ with two conditioned allophones, voiceless [θ] and voiced [ð]), and there are examples of the same word being spelled in some places with þ and in others with ð. For example, Beowulf uses the spellings æþeling and æðeling.

"Old English" comprises at least four main dialect groups that Wikipedia gives as "Mercian, Northumbrian, Kentish, and West Saxon". There were differences in pronunciation between these dialects that were made manifest in various spelling differences.

"Essentials of Old English" (University of Glasgow) mentions the existence of variation between the letters "a" and "o" before nasal consonants.

Not very. There is an idealized normalized spelling used in learners' materials and reference resources like dictionaries, and nowhere else. Heck, people can't even agree on which normalized spelling to use: some prefer the conservative Early West Saxon forms (like hīeran), since they better illustrate the history of the word, while others prefer Late West Saxon forms (hȳran), since Late West Saxon is much better attested.

But both of them exhibit much variation in spelling. The letters þ (thorn) and ð (eth) were completely interchangeable, and many manuscripts used both of them freely. They did tend to use þ at the start of a word and ð at the end, but this was only a tendency, and in the middle of a word, all bets were off. The word oþþe could also be spelled oðþe, oþðe and oððe, and all of these spellings were reasonably common.

The letters a and o were largely interchangeable before the nasal consonants n and m. The word mann can also be encountered as man, monn, or mon. The infinitive form of a verb was usually spelled with -an and the past tense plural form was usually spelled with -on (thus "to love" is lufian and "[they] loved" is lufodon, but less careful writers may still interchange these endings.

For other examples of variation, the word cyning might be encountered in the wild as kyning, kining, cyningc, cynincg, etc. (For some reason, the letter k is much more common in this word than in most other words.)

However, despite all this, Old English spelling is not especially difficult to deal with. If you already know the word cyning, then it'll be pretty obvious what a kiningc is, especially when you see the word in context. There are still underlying rules that explain most of the variant spellings that might occur, and finding a word's normalized form is just a matter of applying those rules in reverse.