Mark has some good suggestions, but I-210 to I-10 in the LA Metro area may not be great if cities bug you. Even 210 can get bogged down unless you drive it at 2 am -- and I wouldn't suggest that at ALL.
Another thought - also long, but few mountains and cities - down the 5 south, take CA-46 over to CA-99 south, then CA=58 east. You will go through some mountains but no cliffs or drop offs, just nice wide 4-lane road. It will drop you on I-15, go north. Follow the signs to I-40 east. That route is all desert, some hills. It's very roll-y, but no cities, cliffs or drop offs. Stay on I-40 east, and the biggest city you'll deal with is Albuquerque. On the interstate, there are certain rules and guidelines that they have to follow, and you will be fine climbing into Kingman, and going through the pass east of ABQ.
On the east side of NM, at Tucumcari, take US-54 through NM, OK's panhandle and into KS. You will be on flatland or at least "rolling hills" by then. I'd personally take the 54 to Pratt (stop by the
World's Largest Hand Dug Well near Greensburg, for a break!) and then head up US-281 to Great Bend, where you can stay on that until you reach I-80 in Nebraska, heading east to Lincoln.
I would also agree with the suggestion to get off the road when you are feeling a bit anxious, and call a reassuring friend. You CAN do this. I might also add that relaxing music might help, something that will calm your body and mind.
Donna