Gibb, the examples in the OED - beginning in the 14th century - all pretty clearly suggest "right away." I think the interpretation of "hastily; without thought" (which I share) may be a 19th or 20th Century development. The OED doesn't mention it.

In the song, it probably just means immediately. That is, right there on shipboard as soon as he thought of it.

There's no way to be absolutely certain because there's not enough context. We don't know what he thought about the marriage later, but if he regretted it the song doesn't say.