I had the same problem last year when I returned to bike commuting. A very busy four lane arterial road for a couple of miles until I got to the bike path where I could relax for the next 6 or 7 miles.
I realized that the problem was that I was thinking like a driver instead of like a rider. I found some residential streets that run effectively parallel to the arterial. Now I have a quite pleasant ride on quiet streets that is just a little bit longer and much less stressful.
So, get out the map and find an alternate.