Damage is by class and by weapon. The simple answer is +1 because that's what's striking.
Describing a second weapon striking is no different than saying "I fill the sky with arrows" (one of the volley 7-9 moves) and hit the enemy with 10 arrows. Do I roll 10 damage? No. You're applying D&D logic to a simpler system that doesn't bother with detailed weapon sims. Why do you do 1d10 and not 2d10 damage? Because damage is by class and your skill at using your weapons. Do you want to fight with a single sword? Groovy. Two handed bastard sword? Groovy. Want to describe yourself wielding a flail? That's cool. How about TWO flails? Also cool. When you start trying to cobble together mechanical bonuses into this is where things start falling apart (but what if I have extra longsword blades on my elbows and knees and describe twirling to slice the enemy up with them?)
+1 or -1 on your to-hit rolls is both boring and very harsh. We're not talking a casual bonus. This is a huge swing in hits, and a more radical shift in full successes.
If you simply can't live without the 1 extra damage, I say go for it. You are after all giving up a shield. But then wonder, what bonus that 2Her gives someone? (Are you going to have to defy danger every time you try to attack someone with a reach weapon?) On the other hand when a sword gets disarmed, broken, melted by dragonsbreath, eaten by a rust monster (take away stuff) you have a spare sword ... so maybe that's a fine deal after all by default.
I say keep it simple. Take your class damage, and take whatever the dominant weapon bonus is. Leave everything else to fictional positioning, and describe yourself fighting however is 'cool' in your mind.