Let's assume Beto O'Rourke wants to run against U.S. Senator John Cornyn in 2020, wins the Democratic senatorial primary and is later nominated as the Democratic vice-presidential candidate.
Is Beto allowed to run for Senator and (Vice) President at the same time?
If so, will there be a special election after Election Day in case Beto wins both the senatorial and the presidential elections, or will Governor Abbott appoint a new U.S. Senator?
If not, how will the new Democratic senatorial nominee be determined?
I know that some states allow candidates to run for two statewide offices at the same time. Joe Biden, for instance, won his senatorial election in 2008 and immediately resigned to become U.S. Vice President. Ted Kaufman was then appointed to continue Biden's term until Chris Coons was elected in a 2010 special election to finish Biden's term.
Are the rules in Texas the same as in Delaware?
Each state has its own rules. Some states like West Virginia actually let legislators also hold a local office, not merely run for two of them. It is not only permitted, it is considered normal.
Texas has a resign to run law that says if you are in the middle of a four-year term for some offices, and run for another office you have resigned the first office.
John Nance Garner was re-elected to Congress and elected to Vice President at the same election. As he had been House Speaker, he went from presiding officer of the House to presiding officer of the Senate on the same day.
In 1952, Allan Shivers was elected governor on both Republican and Democratic tickets. He did this so voters could vote against Harry Truman who was trying to steal Texas oil, and still vote Democratic.
After that, the law was changed that said you could only appear on the ballot once. But then an exception was inserted to permit someone to run for President/Vice President and some other office, this was done so that Lyndon Baines Johnson could run for Vice President and re-election for Senator.
Lloyd Bentsen was re-elected to the Senate in 1988 while also running for Vice-President. Phil Gramm was re-elected to the Senate in 1996, after earlier campaigning for the presidential nomination (the LBJ loophole let him appear twice on the primary ballot).