Article I
Section 4
The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof;
but the Congress may at any time by Law make or alter such Regulations, except as to the Place of Choosing Senators.

Article II
Extracts from Section 1
...
Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or Person holding an Office of Trust or Profit under the United States, shall be appointed an Elector.
...
The Congress may determine the Time of choosing the Electors, and the Day on which they shall give their Votes; which Day shall be the same throughout the
United States.
...

Proposal for Presidential Elections
Each team of Presidential and Vice Presidential candidates on the ballot (as many as care to run from all parties) presents a slate of electors.

The voters of the state rank each team in order of preference. The voter should be free to decide how many preferences to give (unlike the Australian system, which requires a preference for every candidate for a valid ballot paper).

If a team has 50% plus one of the valid votes their slate would be elected. If not the last placed team would be eliminated and any further preferences redistributed. The margin required for victory would be 50% plus one of the ballot papers with a preference for a continuing candidate.

The process would be repeated until a winning ticket emerged.

House and Senate Elections
The same basic system could be used, without needing the complications of a running mate or electoral college members.

Advantages

1. Collapses the primary and general election into one election.
   
2. Eliminates the need for candidate selection conventions.
   
3. Ensures maximum choice for the whole electorate.
   
4. Discourages extremist candidates, who could no longer win just by appealing to a majority of strong partisans and becoming the only choice available to the majority of the electorate from their party.
   
5. Gives a chance for a candidate with wide acceptability to overcome candidates with strong support from a minority of the electorate.

1. Slightly more complex ballot paper. British experience suggests that this is not a major problem for the voter (although professional politicians will not like the added power of the voters, so they would probably allege the change would be too confusing).
   
2. The need for increased precision in election counts. There are more opportunities for the result to turn upon a  small number of votes than in first past the post elections. I concede there might be practical difficulties in the states with larger populations, so the result might take longer to emerge. The question is does the US want an accurate result established over a few days or weeks or a fast one which recent experienced demonstrates may be inaccurate?