Think of that right narrow lane as a bike lane that cars can use. Then ride large as if you belong. Smack dab in the middle of the lane. Trust me, no one will bother you or put you in danger.
Once you do it a few times you won't have any anxiety on that leg. (You do not need to ride fast either! Just take your time.)
The commute will be the best part of your day! Have a great trip, Clay!
Some things bear repeating. CycleDallas team member ChipSeal left this comment in the post below, and it's as good a summation as I can imagine.