2002 Chevy Impala Engine Stutters - Transmission Downshift

My 2002 Chevy Impala is not doing so well as of late. It is a high mileage vehicle (262,000+ Miles), but I am fanatic about maintenance. The engine has recently started stuttering at 35 mph, 45mph, and this morning, even 65mph. It seems to shake like it is trying to down shift, but cannot quite do it. When I step into the gas, the transmission downshift, and the car gets up and goes. But if I am on cruise control at 65 mph, and start going up an incline, the car will shimmer, the tach shows the RPM's jumping between 1500 and 2000, and I can get it to stop by stepping on the gas. I've had catalytic converter replaced (a few times now). Not sure if my issue is engine or transmission related. Any thoughts?