The key here is that you have common bermuda, you really need to mow at 1-1/2". Hybrid bermuda is able to take a lower HOC, but not common, this isn't just my opinion either.
Mowing at 1-1/2" would take care of your thinning problem, help choke out weed's and cut back on water requirement's, you'll just have greener grass. You won't have to do any overseeding, the Sahara you have been using is only a few step's removed from straight common, you won't have any color mis-match's.
Bermuda like's nitrogen, give it 1/2 Lb. of N per 1000 sq. ft. per month, May through Sept. Water the fertilizer in with .25" of water. Mow when the grass is totally dry and keep your blade's sharp. Good luck.