Which is the correct past tense of "spin": "span" or "spun"?
Span is an older strong past tense form of spin — this is the past tense form that existed in the older Germanic ancestors of English. In German, for example, the past tense of spinnen is still spann. In English, span has mostly fallen out of use in favor of spun for both the past participle and simple past forms. This is a form of paradigm leveling. It has occurred in other words like sting (no stang) but not in ring (where we do have rang).
The OED actually gives your quote as follows:
When Adam dalve, and Eve span, Who was than a gentle~man?
Note that this quote is dated from 1560. At this time, span was a more common past tense form, and of course well-known quotes are usually more resistant to being updated to modern vocabulary and grammar. (Interestingly, it looks like dalve has been out of use long enough by now that, at least where you read it, it was updated to delved.)
The OED cites uses of span dating up to the late 19th century, for example, this quote from 1882:
In bad weather she sat at home and span.
So, your answer is: if this sentence were constructed today, then spun would be the correct word, as you suspect. This is simply an old quotation where that word remains fossilized.
The Oxford English Dcitionary entry containing your quote lists both 'span' and 'spun' as spellings of the past tense. In the etymological section however, it notes that [various forms of] 'span' were the singular past tense in both Old and Middle English while the plural past tense was consistently [various forms of] 'spun'/'spon' in Old and Middle English.
The aforementioned quotation was printed in 1560, so doesn't necessarily evidence a tendency either way, and in fact the OED lists
Let theym [wear] suche clothis as they spane.
from around 1450, indicating that perhaps by that time there was no vestige of the quantity distinction that would limit 'spun' to plural usage.
It could also be notable that the latest of the few instances of 'span' in the OED is an 1882-dated usage of it as a transitive verb for spinning material (like wool).
By the way, the quotation appears in the OED as
When Adam dalve, and Eve span, Who was than a gentle~man?
, suggesting that, rare as 'span' was, it may only have been selected in this case for the rhyme.
The conjugation of irregular verbs can vary by dialect. In a very unscientific experiment, I googled "span the thread" and "he span the top", and I got a number of results using span as the past tense of spin, so it seems some people still use span as the past tense of spin, although it's rare. (If you google similarly for uses of stang instead of stung, there are virtually none.)