Do I need a new car radiator?

I have a 2001 Ford Escort with 71,000 miles. A few weeks ago the car overheated on the highway. I had the thermostat replaced and since then the car had been running just fine, however, this has only been for local local trips of no more than a few miles at a time. Two days ago I had to take a long drive (just over an hour on the highway). I watched the temperature gauge carefully the whole time. There were no issues until the end of the trip when the gauge went up very high, about three quarters of the way, (a little too close for comfort). Fortunately we had just about reached our destination when that happened and we were then able to park and let the car cool for about an hour before the drive back home. On the way back, there were again no issues with the temperature gauge right up until the very end when it began to go up to the three-quarter mark. I did use the air conditioner for both drives.

This weekend I have to make a two and a half hour drive and am wondering if I should replace my radiator, or if I should just plan to drive moderately and not use the air conditioner at all. All sincere answers appreciated.

Turn on the AC and see if the radiator fan comes on. It suppose to come on, if not that find out the reason why not. Maybe the fan relay, maybe a switch.

With a cool engine, take the radiator cap off the radiator & see if you can see the cooling tubes in there. Drain some coolant if you have to. If it's nice & clean in there, and there's no crusty build up by the tubes, the radiator should be good to go. If there's build up, replace it.
If you never took the cap off the radiator and you find it was low on coolant now, that's part of the overheating issue you just had.
Also, replace the radiator cap with a new one. An old worn out one won't hold as much pressure and cause overheating.

Lastly, do what Kenny says about the cooling fan. Drive a couple miles with the ac on, then check the fan.