The "small tweaks" version would be to move a skill from under the attribute to the attribute itself; this would mess up the current balance the least, while maybe getting to a more "feels right" direction.
Perc: Could give either lucky looter, salvage or tracker. Each feel like a base perception thing, numbers would need to change of course. I'd go for Tracker, to leave the mats improvements in the skills, but it doesn't fall into 10 easy steps, so something like "range steps" might be needed.
Str: Best would be Pack mule, it feels like the purest base str skill. That will mess a bit with the Pocket Mods, but give the early points one slot per and the later ones 2, that way you'll want a full str build to skip the pockets.
Agi: The stealth perk (sneaking, not damage). Parkour might feel the best for Agi, but not everyone wants it (it'll mess your jumps and possibly your base design)
Fort: Injury resist or Gut improvements; the damage resistance might feel a little strong.
Int: Meds, but this is a tricky one; since most of them are crafting, there's no good candidate. Glorious Leader would be fine, but it's MP only and would require significant rework. Maybe give it something new?
As to tweaking the weapons to correct for the lost headshot damage, make the tweaks emphasize their "style":
Rifles: removed headshot from Perc, increase base headshot damage a bit and add the rest to the rifle perk. This could be used for the default treatment..
Shotties: move toward the mass splatter; remove the extra headshot bonus, but give improved dismemberment at base and at the skill. Blowing off arms and legs is a little pointless, but the overall wreckage would be fun..
Autorifles: improved stun ability... spray and pray a crowd, they'll start toppling on each other
Pistols: recoil reduction (essentially removal), no need to do more per hit if you get less missed heads..
etc..