I have been in this situation numerous times. Here is what I do:
1) Stay calm.
2)Have a backup plan. Know if any other flights are going to where you are going even with another airline.
3) Treat the airline's agent with respect. He/She will probably be yelled at by other passengers so instead, commisserate with that person. I have been known to say "Wow, you're having a tough day aren't you?" (Wait for response). "I hope you can help me." You're commiserating and asking for help. Then suggest your back up plan. You'll be amazed how by just treating them nicely you might get what you want. They have more power than you think.
This doesn't always work but staying calm and treating the people who can help you with compassion and respect is always a better choice.
But the reality is.....while you might get good service in the air, ground personnel are less concerned. They are more worried about on time departures.