I just listened on Google- again - to Driftwood and there isn't much doubt but that he says ' she gave me her paddle'.

The reason it makes sense to me - just barely!- is when I think back to the old phrase of she 'gave me her hand', as in avowing her love and commitment. I have read of a hand being referred to as a paddle, although it was said of a big man. On the other hand the song protagonist is meant to be unpolished, so a paddle may well be fittin'.