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Dueling

When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.

Do I get 1d8+2 instead of straight 1d8 on my Shocking Grasp and Booming Blade ?

One is a melee spell attack, and I want to know if the +2 to damage roll from it applies to melee spell attack.

Shocking Grasp

Lightning springs from your hand to deliver a shock to a creature you
try to touch.
Make a melee spell attack against the target. You have advantage
on the attack roll if the target is wearing armor made of metal. On a
hit, the target takes 1d8 lightning damage, and it can’t take
reactions until the start of its next turn.

The other is a spell that requires you to make a melee attack. Does the attack get +2 to the damage roll as well. Does the 1d8 thunder damage also gets +2 when the target moves?

Booming Blade

As part of the action used to cast this spell, you must make a melee
attack with a weapon against one creature within the spell's range,
otherwise the spell fails.
On a hit, the target suffers the attack's normal effects, and it
becomes sheathed in booming energy until the start of your next turn.
If the target willingly moves be- fore then, it immediately takes
1d8 thunder damage, and the spell ends.

This is confusing to me because after searching left and right I found this answer.

If the melee spell attack is made while the weapon held in hand, does it count for the +2 bonus to damage roll from Dueling Fighting Style?

You can't add Dueling damage to Shocking Grasp but you could add it to the initial attack from Booming Blade

The Dueling fighting style says:

When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.

So, the bonus damage from Dueling applies only when you are making damage rolls with a weapon held in one hand (and not holding another weapon). Do these spells meet that criteria?

Shocking Grasp:

Lightning springs from your hand to deliver a shock to a creature you try to touch. Make a melee spell attack against the target.

This is a melee spell attack with a range of 'touch' and there is no mention that the attack is made via any sort of weapon. You aren't making a weapon damage roll, so you don't gain the Dueling damage bonus.

Your damage for Shocking Grasp is thus the normal 1d8 (unless you have any other reason to add damage to this roll).

Booming Blade:

As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell’s range, otherwise the spell fails. On a hit, the target suffers the attack’s normal effects, and[...]

In this instance you are making a 'melee attack with a weapon' and the target suffers 'the attacks normal effects'. If you have taken the Dueling fighting style (and this attack meets the usual criteria - wielded in one hand, while holding no other weapons) then the 'normal effect' of an attack includes your bonus Dueling damage as part of the damage roll.

Your damage for the initial attack from Booming Blade is thus:

your weapon damage dice (1d8) + your ability modifier (probably strength or dex) + Bonus Dueling damage (2)

Subsequently Booming Blade may deal additional thunder damage but the Dueling bonus is only added once.

When / if the additional thunder damage occurs, it comes from the spell and not the weapon. The bonus that Dueling adds is to damage the weapon deals; the weapon attack is part of the spell, but the secondary damage instance (the thunder damage) isn't from the weapon.

[...] If the target willingly moves before then, it immediately takes 1d8 thunder damage, and the spell ends.