If he got elected to the Senate, does anyone think he would stay from all the pills he was taking? If not, and he stuck around until the 1980's, a drug scandal could really hurt his reelection chances (that's IF he doesn't die from the drug use like OTL).
Also, as a moderate to conservative Southern Democrat, could we possibly see him switch parties after 1994 and become a Republican (that was a popular thing for southern Democrats to do after the "Republican Revolution")
Another thought is his Memphis Mafia. They would no doubt be brought to Washington to keep organized, etc. like they did when he was on tour as a singer. Would the Mafia keep him isolated enough that he would eventually lose touch with the wants and needs of the voters?