Draws

How do you avoid draws with people who continuously check you the same way so that they can avoid losing?

Perpetual check, trying for either a triple repitition of position or the 50 moves (with no pawn move or capure) draw is a very integral part of the game and a very important resource to look for when in a losing position otherwise.

The way to avoid it is to keep your position as secure as possible; though that is sometimes easier said than done. Also, if you must open up your position, try and make sure you will eventually have a way to stop it and still continue your own attacks.

When someone ends up in this situation it's through poor play-- they have allowed their opponent an opportunity to save a losing situation with a draw.  As to how to avoid it, that's very specific to the games, unless you mean something like just being watchful of your own position.  It's somewhat like blundering into a stalemate.  A player knows he is ahead, possibly has a specifc plan for the end, and becomes so intent on executing it that he isn't fully evaluating all of his opponent's options.  Meanwhile, the opponent, who will lose otherwise, is specifically looking for opportunities to force stalemate or perpetual check.  In the dominant position you just have to remember to keep seeing the board through your opponent's eyes too.