No, he didn’t come across as a sore loser, since he didn’t lose. He had an election audaciously stolen from him due to massive fraud and the party establishment colluded to cover up the crime they perpetrated. Real Republicans voted for McDaniel. Democrats had two candidates in November.
That's what a lot of people don't understand. This was a stolen election. Not only that, but the damage caused by the vile tactics that the Barbours used went far beyond Mississippi. They damaged all conservatives by playing up the media lie accusing them of racism.
The Republicans should have burned Cochran after that tactic. Since they didn't, the GOPe is going to keep using it.