Bladesinger dpr and dual wielding

Im in the process of creating a new wizard for a campaign. Looking at bladesinger for higher damage that doesnt consume a bunch of spell slots. I planned to play the character using a concentration spell first round and then weapon attacks. While the bladesinger offers some more damage than other subclasses I dont see it being a huge amount more damage without a dip in a martial or several feats. Not sure if my strategy is correct so im looking for input on how to improve it. I assumed 100% to hit for simplicity.

Looking at level 6 I see combat playing out like this. 

Turn 1. Concentration spell that fits the situation.

Turn 2. Cantrip as one attack = Truestrike (1d6 light weapon attack +d6 radiant damage + 4 (int modifier), then 2nd Melee light weapon attack (d6+4), bonus action attack due to using two light weapons (d6 offhand gets no ability modifier due to lack of two weapon fighting).

Damage per round = 4d6 +4+4 or an average of 14 + 8 =22. Anything more requires the dual wielder feat and/or the nick weapon mastery. This is two feats for a wizard. 

If a concentration control spell isnt feasible. Id use shadow blade or dragons breath to up the damage. Am I forgetting something here or misinterpreting the rules?

The damage looks right to me at first glance, with your best chance of increasing damage is to use Cacphonic Shield or Shadow Blade as the off weapon.

I would also suggest Spirit Shroud as a decent alternative if you just need a good damage increase. It not only adds 1d8 damage to each hit you make, it also reduces all nearby enemies' speed by 10. It's a solid choice if you just need to shell out some damage, and the Bonus Action cast time is nice for getting to still Attack on the first round.