Not sure where you bought your ticket and that may matter. Check the exact change fee penalties for the specific ticket you bought. It's likely $250 + the difference in fare between what you purchased and when you want to fly. If the fare for the later flight is lower, that will be a credit.

Before you do anything, call and find out what the change will cost you in total. Even if the change penalty is waived, if the fare difference is significant, that's another problem.

You don't explain the nature of the emergency, so I will presume that it's a medical emergency respecting an immediate relative. If it's that, your travel insurance may cover the whole thing and you can be done with it.

If it's not, you could try asking UA nicely. They're unlikely to budge, but it's worth a call.

But, be prepared to bite the bullet and pay. The passage of time generally means higher fares.