As I Roved Out is really too common a beginning (or title) to be very helpful. The older folksong collectors reckoned that a good way of getting another few songs out of someone who thought they didn't remember any more was to say "Do you remember that one that starts..." -they almost always did, and it could have been any of quite a large number of completely unrelated songs.

The song in its various forms has been very common throughout the English-speaking world, and persists in tradition to this day under a lot of names. Laws assigned it his category number O17, and it's number 277 in the Roud Folk Song Index.

It was published quite widely on 19th century broadsides, some of which can be seen at the Bodleian Library Broadside Collection, as Nutty has said:

Seventeen Come Sunday

These are mostly of the mid-19th, though the song dates back at least to the late 18th century. There are quite a few previous discussions of it here, which you can find by searching for its various titles through the search engine on the main Forum page. A lot of people are most familiar with the set of which Planxty recorded an arrangement; if I remember right, that was actually a traditional English version, though the song is certainly found in Irish tradition as well.

A previous thread you started on the subject can still be seen at Tune Req: How old are you my pretty little miss