"...Under Arizona law, if McCain were no longer able to serve, the governor must appoint a replacement who has to be from the same party.
The successor would serve until the next statewide election. The winner of that election would serve out the unexpired term, which ends in 2022..."
In Texas, the governor would appoint a temporary senator who would serve till a special election is held. There is no primary. All candidates run in one election. If no candidate wins a majority, the top two candidates face each other in a runoff.