I have never had a problem switching to an earlier flight when mine is showing a delay. When you approach the GA, you need to be concise with your request. For example, "I'm on flight xxx to PHX at 7pm and it's showing a 35 minute delay. Would it be possible for me to get on this flight to PHX?"

Just be very polite and, as long as there are seats available, the GA will let you on the flights.