It looks as if it will be difficult to avoid Interstates 8 and 10 for part of your trip. There don't seem to be many backroad alternatives, but If you have the time, when you reach Benson, AZ, take Highway 80 to the towns of Tombstone (gunfight at the OK corral) and Bisbee (interesting old mining town). Then continue on 80 and either follow it back to I-10 or turn east on Highway 9 (about 15 miles before I-10) and follow it along the Arizona/New Mexico borders with Mexico toward El Paso. If you choose I-10, it will take you through Lordsburg, Deming and Las Cruces, NM, before turning south just west of El Paso. It's a lot of open spaces and 75 to 85 mph speed limits. Though it may not interest you, El Paso is home to the largest Harley dealer in the U.S. (according to the dealer).
From El Paso, I would stay on I-10 heading toward San Antonio until reaching the town of Van Horn (about 200 miles east) then turn Southeast onto Highway 90 and follow it all the way to San Antonio. As you come down the hill into Van Horn, be sure to look toward the right so you don't miss the "Diesel Fried Chicken" sign painted on an old white building. I'm told that's a west Texas delicacy.
Highway 90 will take you through Big Bend country which can be pretty spectacular. You'll pass through the towns of Marfa (home of the "Marfa Lights"), Alpine (close to the McDonald Observatory), and Langtry (home of Judge Roy Bean, Law West of the Pecos). You'll eventually end up in San Antonio. From El Paso to San Antonio on I-10 is about 570 miles. Via Highway 90 it's longer but much more interesting. April will mean temperatures in the 70s to low 80s unless weather trends are abnormal and that's not too bad at all.
Whatever you decide upon, have a safe and enjoyable trip.