I'm handicapped because I can't remember the melody for the verses. It could be that the verses are in C.

However, I've sounded out the tune for the refrain ("When my rowboat comes in...") using your chords. What you have there is a refrain in the key of F.

That might have been a good key for Steve Goodman's voice, but it is not a favorite among guitarists. I'm an amateur, and I don't play in F. the hand positions are just too awkward. But I understand how, when listening to a recording, you want to stick to the same key that you are hearing.

But if you want to stick with F, then a Gm is nice beginning at 'seas.'

The mystery chord for 'sun' is Am.

oth- - F
ther - Gm
side - F
thing - experiment with C and Gm
we go - Gm
rowing - F

Once you've done that, try applying those chords to the verses. If B's remain flat in the verses, they are almost certainly also in the key of F.

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I hope you weren't thinking that only I IV and V chords could be used. The belief that folk songs only use I IV and V chords is just something to tell the children. Kind of the like that the stork brings babies and thunder is the sound of angels bowling.