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?