[jessaustin]

Congratulations on a very manageable LA commute! Having biked many miles for commuting and for pleasure in LA when I lived there, I suspect this is a discomfort you can get over rather than a strictly rational risk assessment. Maybe you could try riding the route at times other than rush hour, like over the weekend, just to get familiar with the route and the basic mechanics of traveling it? Perhaps there is a longer route, over less obvious residential streets, that will be more comfortable? If there is one section that really concerns you, just walk your bike on the sidewalk for that part. There are a number of riding clubs and meetups, with varying degrees of formality and expectations; you might find riding in LA traffic less intimidating with a group. You will find that driving lanes are biking lanes. Swerving from shoulder to shoulder is a mistake you don't have to make.

And I echo sibling's suggestion to get a more practical bike. When you get "off the map" [away from routes that cars are forced to travel] you might find all sorts of sketchy but bike-friendly shortcuts, through parks, parking lots, industrial areas, etc. Skinny road bike tires don't like these routes, but LA streets are full of all the same hazards.