Its summer and lots of traffic through 1-10 tunnel and also the old Bankhead tunnel. I would go North on I-65 then South on I-165 to the alt90 exit. (exit 2 - New Bay Bridge Rd) Its about 6 mi of interstate. The route across that bridge is an industrial area and less traffic and all 4-lane back to I-10.
I have friends in North Mobile and they always use the bridge to avoid heavy traffic in Mobile.
A lot of work has been done on I10 and I12 through Louisiana in the last few years and they are in great shape. A lot of it is now a 6-lane system and the older 4-lane parts have been resurfaced. Currently there is construction from the rest area entering from MS for about 3 mi. then clear sailing to the Texas border. Texas is widening I10 from the LA border for a few miles and this is the last section that needs it. Slow down on this section as they are using the old lanes and they are in bad shape but its only about 6-mi. A good route to Galveston is to take HWY 73 (MM 873) through Port Arthur to Winnie. Its a good 4-lane. This will bypass Beaumont. South out of Winnie on HWY 124 and take the free ferry at Galveston. It a very good ferry and handles 18 wheelers and RV's with no problem.