My part of town is similar with the cul-de-sac residential areas connected by fast arterials. My work commute is over a mile longer on the bike than in the car. A few blocks detour is annoying but won't impact your commute time substantially especially if it makes it easier to safely cross.

I do have some 6- or 8-lane arterials to cross but fortunately most are divided boulevards so I can manage just crossing halfway and then wait until I get another break in traffic. I also structure my commutes so I am not in the rush hour traffic.

Please, do NOT ride upstream against traffic. It is very unsafe. Sidewalks are for pedestrian speeds and are illegal to cycle upon in my area. They are fraught with their own hazards.