The MAF measures the amount of air flowing from the air filter to the engine to determine the correct fuel to air ratio. The two main possibilities I see here are that a) your replacement MAF is defective (which has happened to me), or b) you may have an air leak between the MAF and the intake manifold.
Couple of things I'd try first would be to unplug the MAF and see if your issue goes away. When the engine computer doesn't get a signal from the MAF it defaults to preset readings based on other factors such as RPM and throttle position. It will still run reasonably well, just not quite as efficiently when it has a MAF reporting the actual airflow. If it stops misbehaving with the MAF unplugged, then you've almost certainly gotten a bad part and replacing it (again) should fix the problem.
Otherwise, check the air duct between the MAF and the manifold for any cracks or leaks; if there are it'll be letting in extra air that the MAF wasn't accounting for and throw off your air/fuel ratio.
Give those things a try and let us know what you find.
P.S., Another thing just occurred to me; when my 1999 Beetle's MAF was misbehaving, I found that the plug that connected to the MAF had deteriorated to the point where I wasn't getting good electrical connections, so check your plug for pushed in, broken or missing pins and any signs of corrosion on the pins, along with the wires themselves for breaks or frayed spots.