I was headed in a similar direction. The full ability score bonus would apply to damage, but only part of it to the attack roll.I suggest halving attack bonuses and rounding up so a +1 is still +1 but a +2 is +1 for combat and if you want a +2 you need to have an ability modifier of +3 or +4... The most an 18 <insert ability name> character will get is +3 for its attack and damage rolls.
Str Hit Dam Ability
8 +0 -1 -1
9 +0 +0 -1
10 +0 +0 +0
11 +0 +0 +1
12 +0 +1 +1
13 +1 +1 +1
14 +1 +1 +2
15 +1 +2 +2
16 +2 +2 +2
17 +2 +2 +3
18 +2 +3 +3
19 +3 +3 +3
20 +3 +3 +4