[Note that the interaction darkness-->obscured-->blinded was subject to an erratum; if your PHB is an early printing this answer may look contrary to your rules.]

When you are in darkness someone trying to see you is "effectively blinded" but you are not (PHB p.183, "Vision and Light," errata'ed). Since you are unseen your attack roll has advantage (PHB p.195, "Unseen Attackers and Targets"). But your attack roll against your target--who is lightly obscured--is not affected (PHB p.183, "Vision and Light").

Therefore you attack with advantage. Yay!

(You ask if "they can also see the dimly lit character just fine, and make attacks as normal? " Attacks are normal, but they don't actually see the dimly-lit character "just fine": their perception checks into the dimly-lit area would be disadvantaged, per "Vision and Light.")

Your GM is wrong

The general rule is that, when attacking with a melee weapon, the character can add their Strength modifier (or optionally Dexterity modifier, for Finesse weapons) to damage rolls. From Player's Handbook, page 196, "Damage Rolls":

When attacking with a weapon, you add your ability modifier -- the same modifier used for the attack roll -- to the damage.

From page 194, "Modifiers to the Roll":

The ability modifier used for a melee weapon attack is Strength, and the ability modifier used for a ranged weapon attack is Dexterity. Weapons that have the finesse or thrown property break this rule.

The spell conjures, explicitly, a weapon, but makes no specific exception to the general rule, so the normal logic for attack and damage rolls is used. Since the weapon has the Finesse property, you can and must use either Strength or Dexterity modifier to make the attack and damage roll.

It's also worth noting that all the weapons in the Handbook are described by their damage dice only; for example the greatsword is listed as dealing "2d6" damage, not "2d6 + strength modifier".