Do you have a check-engine light on, or any other lights like the red battery light glowing?
If the alternator were your problem, it wouldn't charge the battery well if at all and you'd eventually have a dead or weak battery. Also, if the alt weren't charging, that battery light on the dash would be on. I'm not saying it's not the alt. The refurbished or replacement alt could be going bad, just that it sounds like your problem is something else. Maybe a failing MAF....
You should go ahead and describe your other problems. If you have a code scanner or know of a local automotive shop (like Autozone) that'll read the codes for free, you should see if the car is giving any codes. Those would tell the forum folks here a lot more about the problems your car might be experiencing.