ZHU Center Controller reported combined sector's complexity increased; without help; resulted in aircraft entering a higher MVA.

Narrative

I was working combined sectors as traffic volume and complexity rapidly increased with an inexperienced supervisor D-side getting his mandatory 8 hours of currency time. The sectors were commanded to be split around the time of the incident which temporarily adds to controller workload and complexity. I had a military aircraft; Aircraft X; on vectors for an approach to Runway 31. He was at the MVA of 022 for his area. The tower changed the runway in use; so I had to vector the aircraft north to set up for an approach to Runway 13. Meanwhile; I get a general aviation aircraft also en route to the same airport. I hadn't decided the sequence yet; but I had time. The two sectors started to increase in volume; and I felt that I was having to do more than I normally would with a D-side due to the less experienced D-side. The GA aircraft advised that he had the airport in site; and I paused a few seconds before answering while trying to calculate whether I could still get Aircraft X to the airport first. Name pointed out that the Aircraft X was on practicing; suggesting that I prioritize the GA aircraft. I decided that would be okay and allowed Aircraft X to continue on a downwind vector. I still thought I would be able to get him onto a base en trail of the preceding aircraft with no problem. Unfortunately; the two sectors continued to get busier. I was late turning the Aircraft X onto base; and he entered an area with a higher MVA of 024 by about 3 miles or so. After I had already turned the aircraft back; Name was very helpful to point out that the aircraft had entered the higher MVA area so that we could get this event reported. The Aircraft X continued to his destination without further incident. It would have been helpful to either have a D-side who works the airspace regularly; split the sectors in a timely fashion; or both.