I got another offer and I'm about to leave my company in one month. The new contract has not been signed yet, so I haven't put in my 2-week notice. There is an extended business trip my team has been planning that takes place a week before my last day. The trip is essential to continue my current projects. Should I go on this trip even though I'm leaving? Or should I defer them until I put in my notice?
Until you put in your notice, you should continue to behave as if you will be remaining at your current company. While the job appears to be in the bag, until you have a signed contract it could still fall though.
As others point out, you should do your job until you leave it. No notice given = haven't left it.
Although, if I worked for a small & underfunded organization, or close friends, I might bend that a little, and look for ways to reduce their expenses without tipping my hand. But if leaving a decent sized company where this travel expense isn't extreme, this is just a normal cost of business to them.
Get on your new employer's case to get the contract signed, put in your two-week's notice and see how events shape up. We have no idea how the management of your current employer is going to react and what decisions they'll make about you including your business trip.
Again, the minute you've got your contract signed - the sooner, the better - give them your two-weeks' notice and let them sort out what to do about you including your business trip. It's less complicated to you that way. And it's the management's job anyway to work this sort of thing out.