It's also worth pointing out that, per the two-weapon-fighting rules, the attacker gets to choose which weapon is being used for a main-hand attack - and this choice can change between attacks. If you have extra attacks, you can alternate between wielded weapons and they all count as main-hand attacks, until you add in the bonus-action off-hand attack, which will use the opposite weapon as one of your main-hand attacks.
Essentially, all 5e characters are ambidextrous. As a DM, I would leave it at that. Losing a hand is punitive enough without adding extra penalties.