The other alternative if you're happy with the existing neck, is to take the fretboard off and put a new one on - something like an ebony one. Not a difficult task, they steam off fairly easily and a new fretboard isn't expensive. You don't need to cut fret slots, so it makes it even easier.

CA glue is another possibility for the fretboard, strings do tend to chew up fretless boards a little, so CA glue gives it some protection.

I'd also sand the neck smaller, definitely the easier option. Apart from anything else it gives you the option of still being able to use the neck it came with if you ever needed to.