Strength gives a bonus "to hit" and "to damage".
Dexterity gives a bonus to "AC" and reaction / initiative.
Have you ever split those bonuses, so instead of a Str Bonus of + 3, instead the player has to choose between bonus to hit and / or damage. So, + hit and no bonus to damage or + 2 hit and + 1 damage.
It might need a slight reworking of the spread of the bonus itself. Just an idle thought that occurred to me.