2001 Hyundai XG 300 - Where to start looking?

After driving for at least 20 minutes this afternoon on surface roads, my Hyundai makes a small lurching motion, like the automatic transmission shifted a little hard. I started heading back home immediately (only about 3 miles away. The shifting/lurching motion got worse as I accelerated & decelerated. Shortly before getting home it seemed as though the electrical & air systems were cycling off & on. I noticed the air especially because it was a very hot day (105 degrees), & it would get completely quiet then loud again. . As I pulled onto my street, it completely died & will not re-start

However, air & electrical seem to be working fine now. Transmission fluid seems ok, as far as i can tell. Manual says to check it while the car is running, & of course I can’t do that now. Fuel was at 1/4 tank. I did note that after driving for awhile the outside temperature reading from my car was showing 110, up from 103 when I first started driving. I don’t know if this could be an indicator that the car was over heating? No warning gauges came on at all. Any suggestions of what to check next would be greatly appreciated.

maybe you have a brolen engane or tranns mount. i would start there.