Part 3 Jason having the axe is fine.
Hell, Part 9 Jason having a double sided axe is actually pretty cool since it would be hard to give Part 9 Jason a real unique weapon beside the dagger or shovel. So fixing his eye to be canon and giving him a unique weapon I have no problems with.
But Part 8 Jason should probably have something besides the axe.
Unless its been confirmed that all the Jasons will be able to choose different default weapons.
So what do you give Jason in Part 8. Maybe the spear gun as seen in the beginning of the movie, but then he wouldn't have any real short range weaponry, unless Part 8 Jason could have the option to shoot the spear gun, with the trade off being a SEVERE cooldown, all the while he can still use it as close range to stab people.