re: UPDATED: Getting people to stay for the whole game -- top ideas in OP

If you are doing a drawing, stretch it out over the season.

At all home games, they will select 3 seats in the 4th quarter (every 5 minutes.) The people in these 3 seats will have until 15 minutes after the game ends to prove that they are present by bringing their ticket stub to an arranged place. Once they have claimed their seat, their name goes into a drawing. At the last home game, you select one name and that person wins one season ticket for the next season. If they are already a season ticket holder, their ticket for next year is free. If they happen to be a fan that is just there for the one game and not a season ticket holder, how happy will they be to win a free season ticket for the next year?

If you really want to make it interesting, do this for all 4 quarters, but a different prize drawing for each quarter. Example: 1st quarter season prize is a free tailgate party for 25 people (BYOB) for one game next year; 2nd quarter season prize is to be on the sidelines for your game of choice next year; 3rd quarter season prize is free parking pass for next year (in a decent lot); 4th quarter season prize is one free season ticket.