If I had a wooden paddle I actually used, I'd prefer the matt finish of oil for the shaft and grip. If you find you don't like it visually, you can always varnish or poly over it.

Linseed oil is mold food but that shouldn't matter for a paddle. Tung oil is normally sold as a wiping varnish. Most are not really an oil finish at that point. Pure tung oil is thicker than motor oil and needs to be thinned with spirits but it will penetrate, fill the pores with wood slurry and harden thereby also hardening the wood, protecting it from dents and the like like no surface coating could. It takes months to truly dry though. I finished a project using pure Tung oil on reclaimed wood so as to preserve the patina and old tool marks that still smells like oil up close over two months later.

That said, my paddle grip is epoxy coated styrofoam and the shaft is sheathed in plastic. I never get blisters but my calluses usually last a good month after a trip and some never seem to go away anymore.

Anyway, you can't go wrong with oil and you can always sand off varnish and poly. Oil will need to be redone every once in a while. Just be careful how you dispose of the rags you use. They can get hot enough to ignite.