2000 Chevy Cavalier Loss of power = problems

Okay so here's my problem. My car starts up fine without any problems. I put the car in drive and start along my way. My Service Light pops on as well as my Theft System light sometimes. Then all of a sudden as I continue to drive on my cars lights and electrical components start to dim or operate at about half of normal i.E. The panel lights are dim when switched to full power, my AC blower slows to almost nothing at full power, my headlights are dimmed, and my windshield wipers go about half as fast as they normally would. When this happens I find that everything will operate normally if I accelerate but as soon as I let off the gas it goes back to half power. Does not stop there. All of a sudden my Check Gauges light will come on and then my heat meter will drop to 0. Sometimes my speedometer, fuel, and RPM meters will also drop to 0 but again will return to normal as I accelerate. Also notice that when I brake my lights will also dim. Finally sometimes my Brake land ABS light will come on, sometimes for a moment and sometimes until I turn my car off. Unfortunately this does not happen all the time. Sometimes none of the above mentioned stuff will occur while I am driving. Other times everything happens. Also another interesting point is that sometimes when the electrical system is running half as well it will return to full power aka will stay completely 100% without me accelerating the engine. IMPORTANT NOTE: I took it into a shop and they said it was the alternator so I had that replaced a little over 12 hours ago and they said that my battery was worn out so they replaced that as well. Literally as I started up my car the same thing happened again, this time with a brand new alternator and battery. PLEASE PLEASE PLEASE help me. After that alternator job I'm down $500+ for services that did NOT fix my problem and am hoping that you will be able to provide me with the right information to avoid anymore unnecessary spending. Thank you very much for your time.