We recently faced an athletic team that applied fullcourt m2m pressure and we lost several possessions in the backcourt and gave up several lay-ups. The strategy we worked on in practice was not utilized in the game once the players got hurried by the pressure. We are as good as anybody except if the opponent is capable of applying heavy, physical pressure.
The most recent pressure defense opponent we played would match-up, trap the dribble and rotate. I align our pressbreak in a 1-2-2 with the 5 inbounding and 1 setting a diagonal backscreen for 2 at the elbow to get open. The 5 would clear out to the front court. We count on them to switch and our 1 pinning her player and getting open for the inbound pass. Once the ball was inbounded and trapped the other guard would rotate behind the trap for a pass, if that pass was unavailable the 5 would flash back to the middle of the court as another option. The 3 and 4 would stay in the front court. I emphasized catching, facing up and looking ahead. I would rather take a 5 or 10 second violation than give up lay-ups.

Is there a better way to teach players to handle fullcourt pressure?

If it's man to man, I usually run a 1-up press breaker (we teach our players how to get open without needing a screen). If their double team is delayed, we clear out. Let the 1 bring it up. If they double right away, it comes from the player guarding the inbounder. So reverse to inbounder and they usually can bring it up. I make sure my inbounds can handle the ball. It's basically 2v2 and if those players aren't skilled enough to bring the ball up in a 2v2 situation, even if they double team, then I know I have a lot of skill work to do with them!

For the most part, I just clear out against man defenses. If it's zone press, I run the typical stuff just keeping good spacing.