We have a place down there on the gulf. I go from Beaumont I10 to Winnie Hwy 124, then drive to the coast. There are campgrounds on Bolivar. Then take the Ferry to Galveston. Avoids all Houston. Have the chance to watch the porpoise and see the ship channel. When you leave Galveston head back north towards Houston and see the sky line while you aim towards your trip to Big Bend.
If you don't want to take the ferry then I'd go Hwy 146 which hooks into I45 south and then you'll next be in Galveston. Again when you leave go north toward Houston to see the skyline. At least these gives you options without worrying about the time you are driving.
Mostly Route 6 is divided. To see the skyline you'll need the main roads downtown like you described, I10, I45 and I69, whichever works on your route.
Just stay out of prime time which is always in downtown Houston so be prepared to stop and no driving in the rain, to much flooding. That being said we've gone right through without any stopping. And do check for progress on road construction on I45 it's horrible. I10 is always under construction in some places with complete stops, Baton Rouge and Beaumont, but it's been a few months since we've been there. These are projects in the works for years though so not an easy fix.