Yes, this is the same family. The ELAMS arrived in Dallas County around 1840. Jesse is the grandfather of my William Curtis Elam, making him my g-g-g grandfather. However, Jesse died in Dallas County in 1854.
The two Jesse ELAMS (15th and 17th TX Cavalry) in the Soldiers and Sailors System are both unknown to me. In fact, they could be the same individual due to the consolidation of the captured/paroled units.