Memory is correct: The Old Maid's Song is in fact on "Hootenanny with the Highwaymen", from early in that group's career, and it's (mostly) the Seeger version...
...except that it turns out that there are a whole lot of performances floating around on YouTube, and the cross-pollination is, well, odd.
First of all, the lead on the Hootenanny version is actually a guest performer, introduced as Mayo Muir (but see notes below).
Second, the recording is notable in that it mostly adopts the lyrics of the Seeger version while applying them to the Kingston Trio arrangement of the tune (again, see notes below). "Mostly" is important: the Hootenanny version is unique in that Muir and the Highwaymen all clearly sing "tinsman" rather than "pinsman" throughout. Now I am curious as to the origin of that choice.
Musically speaking (though I see leeneia's point about the lyrics), I especially like this version; the Highwaymen's madrigal-like backup vocals are extremely deft, considering that according to their intro, Muir was a surprise last-minute addition to the lineup for this live concert recording.
1. Further Googling establishes that she's much better known in the folk world as Ann Mayo Muir, and as 1/3 of a highly distinguished trio consisting of herself, Gordon Bok, and Ed Trickett.
2. The Seeger performances (Pete's here and Peggy's here) are a good deal livelier and less sorrowful than the slower, more mournful pace of the Kingston Trio arrangement. (Note that Pete gives the singer's age as six-and-twenty, whereas Peggy sings "six-and-forty"!)
3. If the upbeat banjo-picking on the Pete Seeger track startles you, consider the even livelier Clive Palmer banjo version from 1967; Palmer's iteration also adopts the novelty of singing both the verse line and the choruses in first person (that is, "I'm a landsman...." (!!)).
4. By contrast, the Glenn Yarbrough version is positively gloomy; like the Highwaymen, he merges what I'm calling the Seeger lyrics with the Kingston arrangement, but with two tweaks to the words: here, the singer is "seven and forty" (thereby the eldest on record) and substitutes "lover" for "offer" in the second verse (!).
5. For completeness, here's a link to the present-day Web site for the original Highwaymen (vs. the newer country-star group, which is a different animal entirely).